A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex...

63
A study of necessary and sufficient conditions for vertex transitive graphs to be Hamiltonian Annelies Heus Master’s thesis under supervision of dr. D. Gijswijt University of Amsterdam, Faculty of Science Korteweg-de Vries Institute for Mathematics Plantage Muidergracht 24, 1019 TV Amsterdam The Netherlands October 2008

Transcript of A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex...

Page 1: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

A study of necessary and sufficient conditions for vertex

transitive graphs to be Hamiltonian

Annelies Heus

Master’s thesis under supervision of

dr. D. Gijswijt

University of Amsterdam,Faculty of Science

Korteweg-de Vries Institute for MathematicsPlantage Muidergracht 24, 1019 TV Amsterdam

The Netherlands

October 2008

Page 2: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part
Page 3: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

Abstract

In this thesis we study some properties of vertex transitive graphs witch are neces-sary for graphs to be Hamiltonian. Next, we also study some sufficiency theorems.Some extra attention will be given to Kneser graphs and we prove Baranyai’s par-tition theorem. We study Cayley graphs and prove all Cayley graphs of abeliangroups are Hamiltonian. In the end we give an overview of further known resultsconcerning our subject.

Page 4: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part
Page 5: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

Contents

Preview 5

1 Vertex transitive graphs 7

1.1 Properties of vertex transitive graphs . . . . . . . . . . . . . . . . . 81.2 Non-Hamiltonian graphs . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Hamiltonicity 17

2.1 Matchings and high degrees . . . . . . . . . . . . . . . . . . . . . . 172.2 Connectivity and toughness . . . . . . . . . . . . . . . . . . . . . . 192.3 Graphs and eigenspaces . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Kneser graphs 25

3.1 Baranyai’s Partition Theorem . . . . . . . . . . . . . . . . . . . . . 263.2 Gray Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3 Proof of Conjecture 3.3 for the case n > 3k . . . . . . . . . . . . . 333.4 The Chvatal-Erdos condition . . . . . . . . . . . . . . . . . . . . . 373.5 The Odd Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.6 The Uniform Subset Graph . . . . . . . . . . . . . . . . . . . . . . 39

3.6.1 The Johnson graph . . . . . . . . . . . . . . . . . . . . . . . 393.6.2 Other Hamiltonian Subset Graphs . . . . . . . . . . . . . . 41

4 Cayley graphs 43

4.1 Abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.1.1 Cycle graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 444.1.2 General abelian groups . . . . . . . . . . . . . . . . . . . . . 45

4.2 The product of Hamiltonian Cayley graphs . . . . . . . . . . . . . 474.3 The Dihedral group . . . . . . . . . . . . . . . . . . . . . . . . . . 484.4 The Symmetric group . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 Prime graphs 55

Epilogue 57

References 59

3

Page 6: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part
Page 7: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

Preview

Definition 0.1. A Hamilton path is a path in a graph G = (V,E) containingevery vertex v ∈ V . A Hamilton cycle is a closed Hamilton path. We call a graphHamiltonian if it contains a Hamilton cycle.

First studied in 1856 and named after Sir William Hamilton, the general problemof finding Hamilton paths and cycles in graphs is already quite old. Althoughthe problem of determining if a graph is Hamiltonian is well known to be NP -complete, there has been a lot of research to the existence of such paths and cyclesin all kind of graphs. There are a lot of examples in which Hamilton paths or cyclesappear. Think about the route of a truck which has to deliver his load in morethan one city. Or the scheduling of more than two soccer teams which all haveto play against each other team. Or even the knight’s tour on a chessboard canbe translated in a Hamilton cycle in a graph. It should be clear why the problemof finding Hamilton cycles in graphs is well known as The Traveling SalesmanProblem.

AB

CD

AC

BD

CE

AD

BE

DE

AE

BC

Figure 1: The schedule of the games of more than two soccer teams, where no team can

play two games in a row and each game is repeated at least once and scheduled in the

same order, is not always easy to find.

5

Page 8: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

The problem of finding Hamilton cycles in graphs can be studied in all kind ofgraphs. To make the problem less broad, in this thesis we will only take a look athighly symmetric graphs, or vertex transitive graphs.

This thesis can be divided into two parts. In the first part we will study thestructure of vertex transitive graphs and see if these graphs meet the conditionsfor graphs to be Hamiltonian. Furthermore we will see some theorems concerningthe Hamiltonicity of graphs. In the second part we will take a close look at somespecial classes of vertex transitive graphs; In particular, Kneser graphs are treatedin Chapter 3 and we will see some Cayley graphs in Chapter 4.We will always restrict our attention to non-directed connected simple graphs and,when not specified otherwise, to k-regular graphs G = (V,E) of order n ≥ 3.For definitions in graph theory, I would like to refer to Introduction to Graph The-ory [39].

6

Page 9: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

Chapter 1

Vertex transitive graphs

Definition 1.1. A graph G = (V,E) is called vertex transitive if for every pairu, v ∈ V there exists an automorphism f : G → G that maps u to v.

Vertex transitive graphs are very well behaved graphs and it seems unsurprisingthat many graphs with such a high symmetry have Hamilton cycles. A bit moresurprising is that there are only five vertex transitive graphs known which do nothave Hamilton cycles, and that there are no graphs known which do not contain aHamilton path. As a consequence, Laszlo Lovasz, born in 1948 in Budapest, camein 1970 with the following conjecture:

Conjecture 1.2 (Lovasz [30]). Every finite connected vertex transitive graph con-tains a Hamilton path.

Another version of Lovasz conjecture states that

Conjecture 1.3. There is only a finite number of vertex transitive graphs whichdo not contain a Hamilton cycle.

It is immediate that every vertex transitive graph is regular. We are going totry to find properties of vertex transitive graphs that are not shared by all regulargraphs. If vertex transitive graphs are more strongly connected than regular graphsin general, it should be easier to find a Hamilton cycle. In this section I will makeintensively use of Algebraic Graph Theory [17].

Let us first introduce some definitions and take a closer look at the structure ofvertex transitive graphs. In this chapter we will see some necessary conditionsfor a graph to be Hamiltonian. We prove that all vertex transitive graphs are1-tough and we will see that all vertex transitive graphs of even order have perfectmatchings. Later in this thesis we will study a couple of specific classes of vertextransitive graphs more intensively.

7

Page 10: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 1. VERTEX TRANSITIVE GRAPHS

1.1 Properties of vertex transitive graphs

Definition 1.4. The vertex connectivity κ(G) is the minimum size of a vertexset S such that G− S is disconnected. If S consists of a single vertex v, we call va cut-vertex. We say that a graph G is k-connected if its connectivity is at leastk.

Definition 1.5. The edge connectivity κ(G) is the minimum size of an edge set Fsuch that G−F is disconnected. If F consists of a single edge e, we call e a bridgeor a cut-edge. We say that a graph G is k-edge-connected if its edge-connectivityis at least k.

Proposition 1.6. If G is a vertex transitive graph of order n ≥ 3, then G doesnot contain a cut-vertex or a bridge.

Proof. (i) Let G be vertex transitive and suppose G contains a cut-vertex. Thenall vertices in G are cut-vertices because G is vertex transitive. Let A1, . . . , Ak bethe components of G − v where v ∈ V and k ≥ 2. We can assume without loss ofgenerality that A1 is the smallest component. Let u ∈ A1. We know that G − ufalls apart in k components B1, . . . , Bk with the components Bi of the same sizesof the Ai. Then for some i it most hold that Bi ⊂ A1 which is a contradiction ofthe fact that A1 is the smallest component.

(i) If G contains a bridge e = {u, v}, and G is of order n ≥ 3, it contains twocut-vertices u and v and G can not be vertex transitive.

Definition 1.7. Let G = (V,E) and A ⊆ V . Denote by NG(A) the neighbors of Anot in A and by A all the vertices not in A and not in NG(A): A = V −A−NG(A).We call A a fragment of G if A 6= ∅ and |NG(A)| = κ(G). A fragment in G ofminimal size is called an atom of G. An edge atom is a subset S ⊆ V such thatdG(S) = κ(G) with S of minimal size.

Definition 1.8. Let N an arbitrary set. A function f : 2N 7→ R is called asubmodular function if for all A,B ⊆ N

f(A ∪ B) + f(A ∩ B) ≤ f(A) + f(B).

Theorem 1.9 (Godsil and Royle, [17]). If G is a k-regular vertex transitive graphwith k ≥ 2, then κ(G) = k.

8

Page 11: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

1.1. PROPERTIES OF VERTEX TRANSITIVE GRAPHS

Proof. It is immediate that κ(G) ≤ k. So we want to prove that κ(G) ≥ k. LetS be an edge atom of G. If S consists of a single vertex then κ(G) = dG(S) = kand we are done. So assume |S| ≥ 2. The graph G is vertex transitive; let f be anautomorphism of G and let T = f(S). Then |T | = |S| and dG(T ) = dG(S). Weknow that dG(A) is a submodular function: for all subsets A,B ⊆ V holds that

dG(A ∪ B) ≤ dG(A) + dG(B) − dG(A ∩ B).

Hence

dG(S ∪ T ) + dG(S ∩ T ) ≤ 2κ(G),

and because S and T are edge atoms this can only hold if either S = T or S∪T = Vor S ∩ T = ∅. Suppose that S ∪ T = V and S ∩ T 6= ∅. Then dG(S) = dG(T\S)and T\S would be a smaller atom than S and T . We may conclude that the edgeatoms of G partition V and for all automorphisms f the following holds:

f(S) = S or f(S) ∩ S = ∅.

Next, suppose that the subgraph of G induced by S is not regular. Let x, y ∈ Swith |NG[S](x)| > |NG[S](y)|. Let g be the automorphism that maps x to y. Then

g(NG[S](x)) = NG[S](y)

and for at least one neighbor z ∈ NG[S](x) of x holds that g(z) ∈ S which contra-dicts the fact that g(S) = S or g(S) ∩ S = ∅.

Hence we know that the subgraph of G induced by S is regular of degree l ≤ kand because G is connected l < k. Let |S| ≥ k, then

κ(G) = dG(S) = |S|(k − l) ≥ |S| ≥ k,

and we are done. Assume next that |S| < k. Since l ≤ |S| − 1, it follows that

κ(G) = dG(S) = |S|(k − l) ≥ |S|(k − |S| + 1) ≥ k,

which proves the theorem.

Theorem 1.10 (Godsil and Royle, [17]). If G is a k-regular vertex transitivegraph, then

2

3(k + 1) ≤ κ(G) ≤ k.

Proof. It is immediate that κ(G) ≤ k. So we want to prove that κ(G) ≥ 23 (k + 1).

Let A be an atom of G. If A consists of a single vertex then κ(G) = |NG(A)| = kand we are done. So assume |A| ≥ 2. The graph G is vertex transitive; let f be an

9

Page 12: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 1. VERTEX TRANSITIVE GRAPHS

Figure 1.1: A 5-regular graph with vertex connectivity four

automorphism of G and let B = f(A). We can check that NG(A) is a submodularfunction so we know

NG(A ∪ B) + NG(A ∩ B) ≤ 2κ(G),

and because A and B are atoms, this can only hold if either

A = B or A ∪ B = V or A ∩ B = ∅,

and, for the same arguments we gave in the proof of Theorem 1.9, we may concludethat V is partitioned by atoms of G.

Let A1 and A2 be atoms. If A1 ∩NG(A2) 6= ∅, then A1 ⊆ NG(A2) otherwise would

NG(A2 ∩ A1) ≤ 2κ(G) − NG(A2 ∪ A1) ≤ κ(G)

and A2∩A1 would be a smaller atom. This tells us that NG(A) is partitioned intoatoms of G, so

|NG(A)| = t|A|

for some integer t. The valency of a vertex in A is equal to k and at most

k ≤ |A| − 1 + |NG(A)| = (t + 1)|A| − 1.

Hence

κ(G) =(t + 1)|A|t

t + 1≥

t

t + 1(k + 1).

Suppose t = 1. Then NG(A) is an atom and |NG(NG(A))| = |A| and since

A ∩ NG(NG(A)) 6= ∅,

it follows that A = NG(NG(A)). But then A = ∅ and by definition A is not afragment. So t ≥ 2 and κ(G) ≥ 2

3(k + 1).

In Figure 1.1, a graph is shown with vertex connectivity smaller than the degreeof a vertex.

In the last two theorems we met a special kind of subset of V .

10

Page 13: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

1.1. PROPERTIES OF VERTEX TRANSITIVE GRAPHS

Definition 1.11. Let G be a vertex transitive graph. A subset S ⊆ V is called ablock of imprimitivity of G if either f(S) = S or f(S) ∩ S = ∅ for all automor-phisms f of G. This partitioning of V is called a system of imprimitivity. A graphwith no nontrivial system of imprimitivity is called primitive.

Note that a system of imprimitivity is nontrivial if there exists a block of imprim-itivity S ⊂ V with 1 < |S| < |V |. We saw that if A is an atom or an edge atom,then A is a block of imprimitivity. Hence if a vertex transitive graph G containsan atom or an edge atom of size bigger then one, G is not primitive.

As we will see later on, the cubic graphs form a set of graphs which are themost critical to be Hamiltonian. We will pay some extra attention to cubic graphsin all sections.

Proposition 1.12 (West, [39]). If G is a cubic graph, then κ(G) = κ(G).

Proof. Let G = (V,E) be a cubic graph and U ⊆ V a vertex cut of size κ(G). Wealready know that κ(G) ≤ κ(G) so it suffices to find a edge cut of size |U |. LetH1,H2 be two components of G−U . Since U is minimal, we know that each v ∈ Uhas at least one neighbor in H1 and at least one neighbor in H2 and because G iscubic, v can not have two neighbors in both H1 and H2. Consider the followingset of edges S:

S = {e ∈ E| v ∈ Uand either e is the only edge from v to H1, or there

are two edges from v to H1 and e is the edge from v to H2}

Then S is an edge cut of size |U |.

With the theory of connectivity and edge-connectivity, we saw how tightly con-nected vertex transitive graphs are. There are two other variables concerning thissubject.

Definition 1.13. An independent set is a set of vertices of a graph G such thatno two vertices in the set are adjacent. The coclique, or independency numberα(G) is the maximum size of an independent set of vertices.

Definition 1.14 (Chvatal, [13]). Let c(G) be the number of components of G. Agraph G is said to be t-tough if t · c(G−S) ≤ |S| for all S ⊆ V with c(G−S) > 1.The largest t such that G is t-tough is called the toughness of a graph and denotedby t(G).

11

Page 14: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 1. VERTEX TRANSITIVE GRAPHS

Unfortunately we do not know much of the size of an independent set of vertextransitive graphs. We do have the following bound

α(G) ≤1

2(|V | + 1),

but this is not a real valuable bound.

We can now prove the following:

Theorem 1.15. All vertex transitive graphs are 1-tough.

Proof. Let G be a k-regular vertex transitive graph. We know by Theorem 1.9 thatG has edge-connectivity κ(G) = k. So there does not exist U ⊆ V with d(U) < k.Let S be a vertex cut. Then

c(G − S)k ≤ dG(S) ≤ |S|k,

so c(G − S) ≤ |S| for all S with c(G − S) > 1.

Concerning the toughness of a graph, note that we can write

t(G) = minS⊆V, c(G−S)>1|S|

c(G − S).

It follows that

t(G) ≤κ(G)

2, and t(G) ≥

κ(G)

α(G).

and κ(G) ≥ 2 if G is vertex transitive.

There are more variants we can look at concerning vertex transitive graphs. Thefollowing theorem shows the relation between vertex transitive graphs and match-ings.

Theorem 1.16 (Godsil and Royle, [17]). Let G be a vertex transitive graph. ThenG has a matching that misses at most one vertex.

Proof. Let G be a vertex transitive graph. A vertex u in G is called critical ifevery maximum matching covers u. It should be clear that if G contains a criticalvertex, then all vertices in G are critical because G is vertex transitive. It followsthat G has a perfect matching if G contains a critical vertex. So assume G containsno critical vertices. Then every vertex in G is missed by at least one matching.Let Mv be a matching of maximum size that misses a vertex v. Suppose thereexists a matching N of maximum size that misses the two vertices u, v ∈ V . LetP be the path between u and v, this path exists because G is connected. We willprove that N does not exist by induction on the length of P .

12

Page 15: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

1.2. NON-HAMILTONIAN GRAPHS

If u and v are connected by an edge, then we could add the edge {u, v} to N ,contradicting the fact that N is maximum. Let the length of P be at least two.Let x be a vertex on P different from u and v. By the induction hypothesis thereexists no maximal matching in G that misses both u and x or both v and x. Hencewe know that N covers x, and that u and v are covered by Mx.Consider the symmetric difference N△Mx. We know that N△Mx has maximumvalency two and each component is either an alternating path or an alternatingcycle, relative to both N and Mx. The vertices u, v, x have degree one in N△Mx.Let Pu be the path with one endpoint in u and Pv the path with one endpointin v. If Pu = Pv, the path is an N -augmenting path with endpoints u and v,contradicting the fact that N is maximal. If the second endpoint of both Pu andPv is equal to x we must conclude that u = v. This concludes the proof that Gcan not have a maximal matching that misses two of its vertices.

1.2 Non-Hamiltonian graphs

There are only five vertex transitive graphs known which do not have Hamiltoncycles. We are very interested in these particular graphs because they can lead us tosome ’rule’ in which cases vertex transitive graphs do not contain Hamilton cycles.The first non-Hamiltonian vertex transitive graph is not interesting at all: it is K2,the graph on two vertices and one edge. The second and third are the Petersengraph on 10 vertices, see Figure 1.2, and the Coxeter graph on 28 vertices depictedin Figure 1.4. We will see both of these graphs in this section. The remainingtwo non-Hamiltonian vertex transitive graphs are obtained from the Petersen andCoxeter graphs by replacing each vertex by a triangle, for picturing this see Figure1.5. This way you get two 3-regular graphs on 30 and 84 vertices respectively. Ifyou repeat the blowing up process a second time the graphs will not longer stayvertex transitive, although they are still 3-regular and non-Hamiltonian.

Ignoring K2, these four non-Hamiltonian graphs are all 3-regular, also called cubic.For this reason we will pay some extra attention to cubic graphs during this thesis.

The Petersen graph

The Petersen graph (see Figure 1.2) is the cubic graph on 10 vertices and 15 edgesand the complement of the line graph of K5. Because the Petersen graph is a smallgraph, it appeared to be a useful example or counterexample for many problems ingraph theory. It is the smallest bridgeless cubic graph with no Hamiltonian cycleand one of the five known non-Hamiltonian vertex transitive graphs.

Theorem 1.17. The Petersen graph does not contain a Hamilton cycle.

Proof. The Petersen graph is constructed from circulants based on Z5 (see Chapter4 for detailed information of graphs based on circulants). Start with the circulants

13

Page 16: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 1. VERTEX TRANSITIVE GRAPHS

Figure 1.2: The Petersen graph and a cycle plus perfect matching. The first is threeedge-colorable, the second can be colored with two colors.

G(Z5, {1}) and G(Z5, {2}) and then add five more edges, joining each one of thesame elements of Z5 in the two circulants. We first show that the Petersen graphis not three edge-colorable. Suppose you can color the edges of the graph withcolors {1, 2, 3}. Without loss of generality we can assume that there are two edgesin G(Z5, {1}) with color 1, two edges with color 2 and one edge with color 3. Nowthe colors for the edges that connect G(Z5, {1}) and G(Z5, {2}) are set. Next wecan not give the edges of G(Z5, {2}) a proper coloring and the Petersen graph isnot three edge-colorable.

Suppose the Petersen graph does have a Hamilton cycle. Then because the graphis three-regular, it is equal to a 10-cycle plus a perfect matching. Color the 10-cyclewith color 1 and 2 in alternating order, and the edges of the perfect matching withcolor 3. Then the graph would be three edge-colorable contradicting the fact thatit was not.

More proofs of the fact that the Petersen graph is non-Hamiltonian are known andwe will refer to this graph a couple of times in this thesis.

The Coxeter graph

The Coxeter graph is the cubic graph on 28 vertices and 42 edges as shown inFigure 1.4 and another vertex transitive graph which is not Hamiltonian. It is

��������

��������

����

����

��������

������������������������������������������������

������������������������������������������������

������������������������������������

������������������������������������

������������

������������

��������������������������������������������������

������������������������������������

������������������������������������

������������������������������������

Figure 1.3: The Fano plane: The antiflags of the Fanoplane are the vertices of the Coxeter

graph.

14

Page 17: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

1.2. NON-HAMILTONIAN GRAPHS

Figure 1.4: The Coxeter graph

constructed from the Fano plane, depicted in Figure 1.3, by drawing a vertex foreach antiflag in the Fano plane. An antiflag is an ordered pair (p, l), where p isa point and l is a line such that p /∈ l. Two such antiflags (p, l) and (q,m) areadjacent in the graph if the set {p, q}∪ l∪m contains all points of the Fano plane.Another way to see this is to make a vertex for each three points which do not lieon a line. Two triplets are adjacent if they are disjoint. This makes the Coxetergraph a subgraph of the Kneser graph K(7, 3) which we will discuss in Chapter 3.

Theorem 1.18 (Godsil e.a. [17]). The Coxeter graph does not contain a Hamiltoncycle.

Different than the non-Hamiltonicity of the Petersen graph, the non existenceof a Hamilton cycle in the Coxeter graph is hard to prove. A direct proof isfound by Tutte [37]. Later in this thesis we will see another way for approachingHamiltonicity, for which we need the following:

Definition 1.19. A graph G = (V,E) is called k-edge regular if for every pair p, qof paths of length k there exists an automorphism that maps p to q.

Theorem 1.20. The Coxeter graph is 3-edge regular.

A proof of this theorem is given by Godsil and Royle [17].

The line graph of the subdivision graph

The non-Hamiltonicity of the two remaining non-Hamiltonian vertex transitivegraphs is not hard to see.

Definition 1.21. The subdivision graph S(G) of a graph G is obtained by puttingone new vertex in the middle of each edge of G. The vertex set of S is equal toV ∪ E, where two vertices v, e are adjacent if v ∈ e in G.

15

Page 18: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 1. VERTEX TRANSITIVE GRAPHS

Figure 1.5: The Petersen graph with each vertex replaced by a triangle

Proposition 1.22. Let G be a cubic graph. Then G is Hamiltonian if and onlyif the line graph of the subdivision graph L(S(G)) is Hamiltonian.

Proof. The line graph of the subdivision graph of a cubic graph G is equal to Gwith each vertex replaced by a triangle, and is itself a cubic graph. The propositionfollows directly from the fact that if a cubic graph on n vertices has a Hamiltoncycle, then it is equal to an n-cycle (plus a perfect matching) and it is easily seenthat a cycle with each vertex replaced by a triangle is still a cycle.

Since we have proved that the Petersen graph and Coxeter graph have no Hamiltoncycles, it follows that the line graphs of the subdivision graphs of these two arenot Hamiltonian either. These are the two remaining non-Hamiltonian vertextransitive graphs we mentioned before. We can repeat the blowing up processinfinitely many times (by taking the line graphs of the subdivision graphs again)and we will get an infinite sequence of non-Hamiltonian graphs. However, thesegraphs will no longer be vertex transitive.

16

Page 19: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

Chapter 2

Hamiltonicity

There are different ways to find out if a graph contains a Hamilton cycle. Wealready saw some necessary conditions for a graph to be Hamiltonian, which vertextransitive graphs meet. Think about not having bridges or cut-vertices, or forcubic graphs to have perfect matchings. We will take a close look at some otherconditions for a graph to be Hamiltonian.

2.1 Matchings and high degrees

In this section we will see that the relation between matchings and cycles is notonly reserved for cubic graphs. Haggkvist conjectured the following theorem whichwas proved by Berman [8].

Theorem 2.1 (Haggkvist). Let G be a k-regular graph of order n ≥ 3 with k ≥12(n + 1). Then every set of independent edges of G lies in a cycle.

The following proposition follows almost directly from this theorem.

Proposition 2.2. Let G be a vertex transitive k- regular graph of order n andk ≥ 1

2(n + 1). Let M be a matching that misses at most one vertex. Then Gcontains a Hamilton cycle C such that the edges of M are contained in C.

Proof. Let G be a vertex transitive graph. First suppose G has an even numberof vertices and let M be a matching of size 1

2n. We saw in Theorem 1.16 that thismatching exists for vertex transitive graphs. Let the degree of the vertices of thegraph be at least 1

2(n + 1). Then it follows directly from Theorem 2.1 that M iscontained in a cycle of size n which is a Hamilton cycle.Suppose next that G has an odd number of vertices and M is a matching of size12(n − 1). Then it follows from Theorem 2.1 that M is contained in a cycle C oflength at least n − 1. If |V (C)| = n, C clearly is a Hamilton cycle and we aredone. So assume |V (C)| = n − 1 and let v be the vertex not in C. Note that Cconsists of edges of M and edges of E(G)\M in alternating order. Assume there

17

Page 20: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 2. HAMILTONICITY

are no two vertices x, y ∈ V (C) ∩ NG(v) such that {x, y} ∈ E(C)\M . Then

dG(v) ≤1

2|M | <

1

2(n + 1)

which is a contradiction. So there are x, y ∈ V (C) ∩ NG(v) such that {x, y} ∈E(C)\M and C\{x, y} ∪ {{x, v}, {v, y}} gives the desired Hamilton cycle.

Dirac slightly improved this bound by

Theorem 2.3 (Dirac, [39]). Let G be a k-regular graph of order n ≥ 3 such that2k ≥ n. Then G is Hamiltonian.

Proof. Let G be a k-regular graph with k ≥ n2 . Assume G contains no Hamilton

cycle. Let G′ be the graph obtained from G by adding a maximum number ofedges without creating a Hamilton cycle. This means that adding any edge to G′

will create a Hamilton cycle.Let u, v ∈ V be two non-adjacent vertices in G′ and let P be the Hamilton path

P = (u = v1, v2 . . . , vn = v).

Define

S = {i : vi+1 ∈ N(u)} and T = {i : vi ∈ N(v)}.

Then S ∩ T 6= ∅ otherwise would S ∪ T = n while n /∈ S ∪ T .Let l ∈ S ∩ T . Then

(u, v2 . . . , vl, v, vn−1, . . . , vl+1, u)

is a Hamilton cycle in G′, a contradiction. So we can not extend G to a maximalgraph G′ and we may conclude that G contains a Hamilton cycle itself.

Jackson [22] at his turn improved this bound by

Theorem 2.4 (Jackson). A k-regular 2-connected graph on n vertices is Hamil-tonian if n ≤ 3k.

We end this section with the following conjecture:

Conjecture 2.5. Let G be a cubic graph. Then there exists a perfect matching Min G such that G − M consists of either one cycle, and G is Hamiltonian, or oftwo non-connected cycles.

We know from Theorem 1.16 that all vertex transitive cubic graphs have perfectmatchings. The only known non-Hamiltonian cubic vertex transitive graphs Ghave perfect matchings M such that G−M consists of two non-connected cycles.If Conjecture 2.5 is true, it is easy to find a Hamilton path and Conjecture 1.2would be true for all cubic graphs.

18

Page 21: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

2.2. CONNECTIVITY AND TOUGHNESS

2.2 Connectivity and toughness

In the previous chapter we studied the connectivity of vertex transitive graphsbecause we expect some relation between the connectivity and the Hamiltonicityof graphs. There exists a theorem stated by Chvatal and Erdos [14] concerningthis relation. For this we first need the following theorem of Dirac [12].

Theorem 2.6. Let G be a graph with κ(G) ≥ 2 and let X be a set of κ(G) verticesof G. Then there exists a cycle in G containing every vertex of X.

With this theorem we can prove the following:

Theorem 2.7 (Chvatal and Erdos). Let G be a graph of order n ≥ 3. If κ(G) ≥α(G) then G is Hamiltonian.

Proof. Let G be a graph with at least three vertices and let κ(G) = s and α(G) ≤ s.Let S = {x1, . . . , xs} be a disconnecting set of order s. According to theorem 2.6,every subset U ⊆ V with |U | = s is contained in a cycle in G. Let C be thelongest cycle containing the vertices of S. Assume C is not a Hamilton cycle andlet v /∈ C. According to Menger’s theorem there exist pairwise disjoint paths pi,such that pi goes from v to xi, where i = 1, . . . , s. Fix an orientation for C andlet yi be the predecessor of xi in C.

i) Assume there are i, j ∈ {1, . . . , s} with xi = yj, then we could replace the edge{yj , xj} by the path (pi, pj) and obtain a longer cycle, which is a contradiction.So xi 6= yj, i, j = 1, . . . , s.

ii) Assume there is an i ∈ {1, . . . , s} with yi adjacent to x, then we could replacethe edge {yi, vi} by the path (yi, v, pi) and obtain a longer cycle,which is a contra-diction. So yi is not adjacent to v for all i ∈ {1, . . . , s}.

iii) There are a, b ∈ {1, . . . , s} for which the edge {ya, yb} exists in G. If not,{y1, . . . , ys, v} form an independent set of size s + 1, which is a contradiction.

Now we could delete the edges {ya, xa} and {yb, xb} from C and replace them by

C

v

ya

xa

xb

yb

pa

pb

Figure 2.1: A cycle including the vertices of an independent set

19

Page 22: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 2. HAMILTONICITY

{ya, yb} and (pb, pa) and obtain a longer cycle (see Figure 2.1), which is a contra-diction. We may conclude that C is a Hamilton cycle.

As a result of this theorem we give the following proposition without proof.

Proposition 2.8 (Chvatal and Erdos [14]). Let G be a graph of order n ≥ 3. Ifκ(G) + 1 ≥ α(G) then G has a Hamilton path.

We already know some bounds for κ(G) for k-regular vertex transitive graphs:

2

3(k + 1) ≤ κ(G) ≤ k.

If we can say something about the bounds of α(G), we know how useful Theorem2.7 is. Without any further knowledge of G, the only thing we can say aboutvertex transitive graphs on n vertices is that α(G) ≤ 1

2(n + 1). So if

1

2(n + 1) ≤

2

3(k + 1),

then G is Hamiltonian, which really is not a good bound: We have seen betterbounds by different theorems before.

Another necessary condition for a graph to be Hamiltonian, is to be 1-tough.We already saw that vertex transitive graphs meet this constraint. It is a interest-ing question if there exists a t such that every t-tough (vertex transitive) graph isHamiltonian. Chvatal [13] stated the following conjecture.

Conjecture 2.9. There exists t0 such that every t0-tough graph is Hamiltonian.

This conjecture is valid for planar graphs and t0 > 3/2: Every t-tough graph witht > 3

2 is 4-connected and by Tutte’s theorem [38], every 4-connected planar graphis Hamiltonian. Unfortunately this is not relevant in order of our study of vertextransitive graphs because most vertex transitive graphs will not be planar. Thetoughness of the Petersen graph is equal to 4

3 and the toughness of the Coxetergraph is equal to 3

2 . In fact, there are no counterexamples known of the following:

Conjecture 2.10. Every 2-tough graph is Hamiltonian.

20

Page 23: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

2.3. GRAPHS AND EIGENSPACES

2.3 Graphs and eigenspaces

Let G = (V,E) be a graph and AG be the V × V matrix with entrees aij whereaij = 1 if {i, j} ∈ E and 0 otherwise. We call AG the adjacency matrix ofG. Further let LG = DG − AG be the Laplacian matrix of G, where DG is thediagonal matrix with the degrees of the vertices of G on its diagonal. In thisthesis we only look at vertex transitive graphs so we can write the Laplacian asLG = kI − AG, where k is the degree of the vertices of G and I is the identitymatrix of appropriate dimension. There has been a lot of study on the eigenvaluesλi of a graph, i = 1 . . . n. For a survey of results, see for instance Mohar [35].By the eigenvalues of a graph we mean the eigenvalues of the adjacency matrixAG. We will order the eigenvalues such that λ1 ≤ λ2 ≤ . . . ≤ λn. The set of alleigenvalues of a graph G is called the spectrum spec(G) of a graph G.Of course you can also take a look at the eigenvalues of the Laplacian matrixLG. To avoid misunderstanding we shall write λi(AG) and λi(LG) for the i-theigenvalue of the adjacency matrix and the Laplacian matrix respectively. Notethat the eigenvalues of the Laplacian are ordered such that λ1(LG) is the largesteigenvalue and:

λi(LG) = k − λi(AG) (2.1)

In this section we will see that we have a lower bound for the Laplacian eigenvaluesof a graph G, for G to be Hamiltonian (Heuvel [19]). We first choose an orientationon G by choosing for every edge e of G one of its end vertices as the initial vertexand the other as the terminal vertex. Define the oriented incidence matrix of Gwith respect to the chosen orientation by the V × E matrix MG with entries

(M(G))ve =

1, if v is the terminal vertex of e,

−1, if v is the initial vertex of e,

0, if v and e are not incident.

Let MTG denotes the transpose of MG. It holds that LG = MG · MT

G .Both MG ·MT

G and MTG ·MG are symmetric matrices with only nonnegative eigen-

values and, apart from the multiplicity of the eigenvalue 0, their spectra coincide.So

spec(MTG · MG) = {spec(MG · MT

G), 0[m−n], }

where n is the number of vertices of G and m is the number of edges of G. Let Hbe a subgraph of G obtained by deleting an edge of G. Then the matrix MT

H ·MH

is obtained from MTG · MG by deleting the corresponding row and column of the

deleted edge (MTG · MG is an E × E matrix). By the Interlacing Property of

symmetric matrices (van Lint [29]) we now have

λi(MTG · MG) ≥ λi(M

TH · MH) ≥ λi+1(M

TG · MG), i = 1 . . . n − 1.

21

Page 24: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 2. HAMILTONICITY

It follows that

λi(LG) ≥ λi(LH) ≥ λi+1(LG), i = 1 . . . n − 1.

If you delete m′ edges from G this results in

λi(LG) ≥ λi(LH) ≥ λi+m′(LG), i + m′ ≤ n.

Theorem 2.11 (Heuvel, [19]). Let G be a graph on n vertices and λi(LG), i =1 . . . n its Laplacian eigenvalues. If G contains a Hamilton cycle then

λi(LG) ≥ λi(LCn) ≥

{

λm−n+i(LG) if 1 ≤ i ≤ 2n − m0 otherwise.

Proof. If G contains a Hamilton cycle Cn, you can delete m′ = m − n edges fromG to contain a subgraph H equal to Cn. Now we have

λi(LG) ≥ λi(LH) ≥ λi+m−n(LG), i + m − n ≤ n.

It follows directly from Theorem 2.11 and (2.1) that for k-regular graphs on nvertices it holds that

λi(AG) ≤ λi(ACn) + (k − 2).

We see for the Petersen graph P with spectrum spec{−2[4], 1[5], 3[1]} that

λ5(AP ) = 1 > 2cos(3π/5) + 1 = λ5(AC10) + 1

which makes the Petersen graph non-Hamiltonian.

We now take a closer look at the Coxeter graph C. Suppose C contains a Hamil-ton cycle H. As a result of Theorem 1.20, we can choose any path of length threewhich should be contained in H. Lets say we choose the path

P = (vi, vi+1, vi+2, vi+3) ⊂ H.

Each vertex vj of C is contained in three edges; let ej be the edge with vj ∈ ej

and ej /∈ H. LetC ′ = C\{ei+1, ei+2}. (2.2)

Then it should hold that C ′ still contains the cycle H. In Table 2.1, the Laplacianeigenvalues of C28, the Coxeter graph and the Coxeter graph minus the two non-adjacent edges as in (2.2) are shown. Unfortunately, the Interlacing Propertystill holds for C ′, and we can not exclude the Coxeter graph to be Hamiltonian.Nevertheless, it is a good example of deleting edges from graphs and comparingtheir eigenvalues.

22

Page 25: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

2.3. GRAPHS AND EIGENSPACES

λi C28 C C ′ λi C28 C C ′

λ1 4 5.414 5.414 λ15 2 2.586 2.586λ2 3.950 5.414 5.414 λ16 1.555 2.586 2.586λ3 3.950 5.414 5.414 λ17 1.555 2.586 2.586λ4 3.802 5.414 5.414 λ18 1.132 2.586 2.215λ5 3.802 5.414 5.228 λ19 1.132 2.586 1.864λ6 3.564 5.414 4.751 λ20 0.753 1 1λ7 3.564 4 4 λ21 0.753 1 1λ8 3.247 4 4 λ22 0.436 1 1λ9 2.247 4 4 λ23 0.436 1 1λ10 2.868 4 4 λ24 0.198 1 1λ11 2.868 4 4 λ25 0.198 1 1λ12 2.445 4 3.464 λ26 0.050 1 0.716λ13 2.445 4 3.318 λ27 0.050 1 0.444λ14 2 2.586 2.586 λ28 0 0 0

Table 2.1: the Laplacian eigenvalues of C28, the Coxeter graph C and the Coxeter graphminus the two edges C′ as in (2.2).

23

Page 26: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part
Page 27: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

Chapter 3

Kneser graphs

Definition 3.1. A k-subset of a set N is a subset of N with cardinality k. TheKneser graph K(n, k) has as vertices the k-subsets of {1, 2, . . . , n} where two ver-tices are adjacent if the k-subsets are disjoint.

Kneser graphs are named after Martin Kneser, who first investigated them in 1955.The graph in the introduction of this thesis is the Kneser graph K(5, 2). This graphis also equal to the Petersen graph, so we know that there is no solution for thesoccer problem described. Though for most pairs (n, k) the use of Kneser graphsis a good way to find a schedule for the matches to play, in fact for all n ≥ 6there exist solutions of this scheduling problem as we will prove in this section. Ofcourse you can find many other applications comparable to this soccer problem.

Theorem 3.2. All Kneser graphs are vertex transitive.

Proof. Let f be some permutation of {1, 2, . . . , n}. Then f induces an automor-phism of K(n, k): Let S = {a1, . . . , ak} and T = {b1, . . . , bk} be vertices of K(n, k).Then, for example,

(a1b1)(a2b2) · · · (akbk) (3.1)

is a permutation of {1, 2, . . . , n} that sends S to T and the neighbors of S to theneighbors of T . So (3.1) is an automorphism of K(n, k) and all Kneser graphsare vertex transitive. Note that this is only an example of the many permutationsthat map S onto T .

As a result of Conjecture 1.3 we now have the following.

Conjecture 3.3. All Kneser graphs are Hamiltonian, except for K(5, 2), whichis the Petersen graph.

It is already shown for a lot of Kneser graphs that they are Hamiltonian. Inparticular, K(n, k) has a Hamilton cycle for k = 1 and n ≥ 3, which is thecomplete graph Kn. Heinrich and Wallis [18] showed the Hamiltonicity for k = 2and n ≥ 6 and for k = 3 and n ≥ 7. Chen [10] constructed a Hamilton cycle for

25

Page 28: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 3. KNESER GRAPHS

n ≥ 3k and even improved this bound later on by n ≥ 2, 62k + 1. Note that fora graph to be connected it is necessary that n ≥ 2k + 1 which still gives us a gapfor Kneser graphs to be Hamiltonian when 2k + 1 ≤ n ≤ 2, 62k + 1.In this section we will study some of the results known. We prove an importanttheorem of Baranyai, we introduce Gray codes, and prove they exist.

3.1 Baranyai’s Partition Theorem

Definition 3.4. A hypergraph H is a pair (V,E) where V is a set of vertices andE is a set of subsets of V . The k-uniform hypergraph Kk

n has n vertices and allthe k-subsets of {1, 2, . . . , n} as edges.

The degree d(v) of a vertex v of a hypergraph is defined as the number of edges Ewith v ∈ E. We call a hypergraph almost regular if |d(u) − d(v)| ≤ 1 for all pairsu, v ∈ V . A famous theorem on partitioning the edges of a hypergraph is statedby Baranyai.

Theorem 3.5 (Baranyai’s partition theorem). Let a1, . . . , at be positive inte-gers with

∑ti=1 ai =

(

nk

)

. Then the edges of Kkn can be partitioned into almost

regular hypergraphs Hi = (V,Ai) with vertex set V = {1, . . . , n} and edge setAi = {Ei

1, . . . , Eiai}, for i = 1 . . . t.

Schrijver and Brouwer [29] proved this theorem for the case that all ai are equaland n = sk for s ∈ N. We will extend this proof to the general case. In our proofwe will use the following theorem:

Theorem 3.6 (Integer flow theorem). Let D = (V,A) be a directed graph and lets, t ∈ V . Let f : A → R+ be an s − t flow of value β. Then there exists an s − tflow fint : A → Z+ of value ⌊β⌋ or of value ⌈β⌉ such that

⌊f(a)⌋ ≤ fint(a) ≤ ⌈f(a)⌉

for each arc a ∈ A.

Proof. We are going to prove by induction a somewhat stronger statement. Leta1, . . . , at be positive integers with

∑ti=1 ai =

(

nk

)

. We assert that for any l =0, . . . , n, there exist families of subsets of {1, . . . , l}

A1, A2, . . . , At

such that |Ai| = ai and each subset S ⊆ {1, . . . , l} is contained in exactly(

n−lk−|S|

)

of the Ai. To be sure that exactly ai subsets are contained in every Ai, we allowsubsets to occur with a multiplicity in every Ai. Notice that

(

n − l

k − |S|

)

= 0 if |S| > k.

26

Page 29: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

3.1. BARANYAI’S PARTITION THEOREM

σ τ

A1

Am

S

f=k−|S|−n−l−

Ø

{1,..., l}

i −ΣS∈ Ai f=f=ka |S|

n−l

n−lk−|S|

11

Figure 3.1: A (σ − τ)-flow through the bipartite graph B

If we take l = 0, these families indeed exist; each Ai will consist of ai copies of ∅.

Assume for some l < n families with the desired properties exist. We now canconstruct a bipartite graph G with vertex sets

U = {A1, A2, . . . , At}, W = 2{1,...,l},

so W consists of all subsets of {1, . . . , l}. A vertex Ai is connected to a vertexS ∈ W if S ∈Ai. If S occurs in Ai with multiplicity a, then we make a paralleledges between Ai and S. Then G has the following degrees:

dG(Ai) = ai, dG(S) =

(

n − l

k − |S|

)

.

Direct every edge from U to W . Let σ be a source and connect σ to the verticesof U . Let τ be a sink and connect τ to the vertices of W . See Figure 3.1 for theconstruction of this graph. Let f be the flow from σ to τ where the flow valueof the edges leaving σ, the flow value of the edges from Ai to each of its mem-bers S and the flow value of the edges from a subset S to τ are respectively equal to

f(σ,Ai) =kai −

S∈Ai|S|

n − l, f(Ai,S) =

k − |S|

n − l, f(S,τ) =

(

n − l − 1

k − |S| − 1

)

.

This indeed is a flow:

a∈dout(Ai)

f(a) =∑

S∈Ai

k − |S|

n − l=

1

n − l(aik −

S∈Ai

|S|) =∑

a∈din(Ai)

f(a), and

27

Page 30: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 3. KNESER GRAPHS

a∈din(S)

f(a) = din(S)k − |S|

n − l=

k − |S|

n − l

(

n − l

k − |S|

)

=

(

n − l − 1

k − |S| − 1

)

(3.2)

=∑

a∈dout(S)

f(a).

It follows because of the Integer flow theorem that there also exists an integer flowfint of the same value for this network with

⌊f(a)⌋ ≤ fint(a) ≤ ⌈f(a)⌉

for all arcs a in G. Let hl = aik −∑

S∈Ai|S|. All edges leaving σ will get some

flow value

ki = fint(σ,Ai) ∈ {

hl

n − l

,

hl

n − l

}

under fint. Furthermore

fint(Ai,S) ∈ {

k − |S|

n − l

,

k − |S|

n − l

},

and the outflow of a vertex S ∈ W remains the same under fint.

We are now going to add the extra element l + 1. Denote by Sij those membersof Ai with

fint(Ai,Sij) =

k − |Sij |

n − l

,

where j = 1, . . . m with m ≤ ki. Note that it is possible that there are sets Sij

equal to each other. Next let

Tij = Sij ∪ {l + 1}. (3.3)

We first prove by induction to l that it follows from this construction that

k − |S|

n − l≤ 1 for all l and for all S ⊆ {1, . . . , l}. (3.4)

For l = 0 this follows from the fact that k ≤ n. Assume (3.4) holds for some l ≤ n.If it even holds that

k − |S| < n − l for all S ⊆ {1, . . . , l},

then it is immediate that also

k − |T | ≤ n − (l + 1) for all T ⊆ {1, . . . , l + 1}.

Suppose for some S ⊆ {1, . . . , l} equality holds in (3.4). Then

k − |S|

n − l

=k − |S|

n − l= 1

28

Page 31: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

3.1. BARANYAI’S PARTITION THEOREM

and according the construction in (3.3), the element l + 1 is added to the subsetS and it still holds that

k − (|S| + 1)

n − (l + 1)= 1

and we have proved (3.4).

We may conclude that fint assigns the value 1 to ki of the edges leaving Ai and 0to the others. It follows from (3.2) that

a∈δin(S)

fint(a) =∑

i:fint(Ai,S)=1

1 =

(

n − l − 1

k − |S| − 1

)

, for all S,

so for each S the set S ∪ {l + 1} will be covered by(

n−l−1k−|S|−1

)

of the Ai, includingmultiplicity.

Define

Bi = Ai\{Sij}ki

j=1 ∪ {Tij}ki

j=1.

Let S ⊆ {1, . . . , l} and T ⊆ {1, . . . , l +1}. We distinguish two cases: l+1 /∈ T andT is contained in

(

n − l

k − |S|

)

(

n − l − 1

k − |S| − 1

)

=

(

n − l − 1

k − |T |

)

of the Bi, or l + 1 ∈ T and T is contained in

(

n − l − 1

k − |S| − 1

)

=

(

n − l − 1

k − |T |

)

of the Bi. We see that it holds that there are ai subsets T ⊆ {1, . . . , l+1} in each Bi

and each subset T is contained in exactly(

n−l−1k−|T |

)

of the Bi. We can conclude that

B1, B2, . . . , Bt are families of subsets of {1, . . . , l + 1} with the desired properties.

If we take l = n we have

(

0

k − |S|

)

=

{

1 if |S| = k,0 otherwise.

so all subsets S are k-subsets and we have found a partition with ai k-subsets ineach Ai and because

∑ti=1 ai =

(

nk

)

, all k-subsets of {1, . . . , n} will meet exactly onehypergraph Ai. Next, we only have to prove that this partition gives us almostregular hypergraphs Hi = (V,Ai). We saw in our induction step that any newelement l is covered by Ai exactly ki times with

ki ∈ {

hl

n − l

,

hl

n − l

}

29

Page 32: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 3. KNESER GRAPHS

and hl = aik −∑

S∈Ai|S|. If we can prove that ki ∈ {

aikn

,⌈

aikn

} for all l, then

we have proved that the number of times every element of {1, . . . , n} appears inAi differs at most one.

Proof by induction to l. For l = 0 this is easy to see:

ki ∈ {

h0

n

,

h0

n

} = {

aik

n

,

aik

n

}.

Let for some l < n, ki ∈ {⌊

aikn

,⌈

aikn

}. We know that for l + 1 holds that

ki ∈ {⌊

hl+1

n−l−1

,⌈

hl+1

n−l−1

}. It is not difficult to see that

hl+1 ∈ {

hl(n − l − 1)

n − l

,

hl(n − l − 1)

n − l

}.

It follows that

hl+1 ≥(n − l − 1)(n − l)⌊aik

n⌋

n − land hl+1 ≤

(n − l − 1)(n − l)⌈aikn⌉

n − l.

Hence, for all l ki ∈ {⌊

aikn

,⌈

aikn

}.

3.2 Gray Codes

The term combinatorial Gray code was introduced in 1980 to refer to any listing ofcombinatorial objects so that successive objects differ in some pre-specified smallway. This notion generalizes the classical reflected binary code for listing n-bitbinary numbers such that successive numbers differ in exactly one bit position.The reflected binary code was introduced by Frank Gray in 1947. For now, we willwork with the following definition:

Definition 3.7. Let m =(

ab

)

. Let us denote a set {1, ..., a} by [a] and the set of

all b-subsets of a by([a]

b

)

. A Gray code G(a, b) is a list (D1, . . . ,Dm) of all the

members of([a]

b

)

such that |Di ∩ Di+1| = |Dm ∩ D1| = b − 1 for i = 1 . . . m.

Gray codes can be used for error correction in digital communications as for manylistings in combinatorics, such as listing permutations or combinations. Gray codesare also used in many proofs involving Hamilton cycles.

Theorem 3.8. Gray Codes exist for all a ≥ b ≥ 1.

30

Page 33: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

3.2. GRAY CODES

A

A

i

i∪

n+1

n+1

+1}n{

u

v’ u’

vD

E

A

Az y

z x

y

x

Figure 3.2: The Gray code G(n + 1, b)

Proof. We are going to prove the slightly stronger statement that for all a ≥ b ≥1 and for any pair of b-sets u, v with u ∩ v = b − 1, there exists a Gray code(D1, . . . ,Dm) with u = Di, v = Di+1. Proof by induction to a. It is easy to checkthat for a = 1, 2, 3 and all 1 ≤ b ≤ a and for any pair of b-sets, Gray codes existwith the desired properties. Assume for a = n ≥ 3 and all 1 ≤ b ≤ a and for anypair of b-sets there exists a Gray code G(a, b) with the desired properties. Take

a = n + 1. The members of([n+1]

b

)

can be written as

(

[n + 1]

b

)

=

(

[n]

b

)

((

[n]

b − 1

)

⊕ {n + 1}

)

,

where A ⊕ {e} means that the element e is added to every element of the set A.Let

v ∈

(

[n]

b − 1

)

⊕ {n + 1} and u = v\{n + 1} ∪ {z}, z ∈ [n].

We are going to prove that there exists a Gray code G(n + 1, b) such that {u, v} issubject of G(n + 1, b).Let x ∈ v and

v = v\{x} ∪ {y}, and u = u\{x} ∪ {y}, for some y /∈ v.

We know, because of the induction hypothesis, that there exist Gray codes

(D1, . . . , u, u, . . . ,Dm)

of([n]

b

)

and

(E1, . . . , v\{n + 1}, v\{n + 1}, . . . , El)

of( [n]b−1

)

, where m =(

nb

)

and l =(

nb−1

)

. Define Ei = Ei ⊕ {n + 1} for i = 1, . . . , l.Notice that

(E1, v, v, . . . , El)

31

Page 34: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 3. KNESER GRAPHS

is a Gray code of( [n]b−1

)

⊕ {n + 1}. Then, because u = v\{n + 1} ∪ {z},

(v = Ej , . . . , El, E1, . . . , Ej−1 = v, u = Di−1,Di−2, . . . ,D1,Dm, . . . ,Di = u)

is a Gray code of([n+1]

b

)

. For picturing this, see Figure 3.2

Notice that if we would do the same for |Di∩Di+1| = |Dm∩D1| = b−k, it doesn’twork because we get for k = 2, a = 5 and b = 2, 3 a system inducing the Petersengraph, and for k > 2 we can not construct a list such as in Figure 3.2.

We can now prove the following theorem of Chen and Furedi [9].

Theorem 3.9 (Chen, [9]). The Kneser graph K(n, k) is Hamiltonian for n = sk,where 3 ≤ s ∈ N.

Proof. Let n = sk, 3 ≤ s ∈ N and m =(

n−1k−1

)

. Denote a k-set of [n] by Eij . By

Baranyai’s partition theorem there exists a partition

(

[n]

k

)

=

m⋃

i=1

{Ei1, . . . , E

is},

such that

Eij ∩ Ei

k = ∅, and

s⋃

j=1

Eij = [n].

Without loss of generality we may assume that {n} ∈ Ei1 for all i. Let Di =

Ei1\{n}. Note that

m⋃

i=1

Di =

(

[n − 1]

k − 1

)

.

Then (permute if necessary) {Di}mi=1 forms a Gray code G(n − 1, k − 1). Let

zi = Di+1\Di and zm = D1\Dm. We can assume that (permute if necessary)zi /∈ Ei

s. Note that it is important that s ≥ 3. Then Ei+11 ⊂ Ei

1 ∪ {zi} andEi+1

1 ∩ Eis = ∅ (and E1

1 ⊂ Em1 ∪ {zm} and E1

1 ∩ Ems = ∅). Now

E11 , . . . , E1

s , E21 , . . . , E2

s , . . . , Em1 , . . . , Em

s

form a Hamilton cycle through K(n, k). See Figure 3.3

32

Page 35: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

3.3. PROOF OF CONJECTURE 3.3 FOR THE CASE N > 3K

1E21Em

2E1

sE1sEm

Ej1

E11

Figure 3.3: A Hamilton cycle through K(sk, k). Note that Eij ∩ Ei

k = ∅, and Ei+1

1 ⊂

Ei1 ∪ zi, zi /∈ Ei

s and s ≥ 3.

3.3 Proof of Conjecture 3.3 for the case n > 3k

We prove Conjecture 3.3 for the case n > 3k by induction on k. The proof is dueto Chen [10].

For k = 1 the Kneser graph K(n, 1) is equal to the complete graph and obvi-ously has a Hamilton cycle for n > 3. Induction Hypothesis: The Kneser graphK(n, k − 1) has a Hamilton cycle for all n > 3(k − 1).

Let n > 3k. We are going to prove that K(n, k) has a Hamilton cycle. We will dothis by building a Hamilton cycle through the vertices

V =

(

[n]

k

)

= V0 ∪ V1 ∪ V2 ∪ V3, with

V0 =

(

[n − 2]

k

)

, V1 =

(

[n − 2]

k − 1

)

⊕ {n − 1},

V2 =

(

[n − 2]

k − 1

)

⊕ {n}, and V3 =

(

[n − 2]

k − 2

)

⊕ {n − 1, n},

where the Vi are disjoint.

Let us construct a system where we can use Baranyai’s partition theorem. Letn − 1 = sk + r, where 3 ≤ s ∈ N and 0 ≤ r ≤ k − 1. Further, let

(

n − 1

k

)

= s(t − 1) + q, where 1 ≤ q ≤ s and let m =

(

n − 2

k − 1

)

.

Note that

s(t − 1) + q =

(

n − 1

k

)

=

(

sk + r

k

)

≥ s

(

sk + r − 1

k − 1

)

= s

(

n − 2

k − 1

)

= sm, (3.5)

33

Page 36: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 3. KNESER GRAPHS

whence t ≥ m. Define

ai = s for i ≤ t − 2at−1 = s, at = q if q ≥ 2at−1 = s − 1, at = 2 if q = 1.

(3.6)

Thent

i=1

ai =

(

n − 1

k

)

= |E(Kkn−1)|,

and by Baranyai’s theorem the edge set E(Kkn−1) of Kk

n−1 can be partitioned intot almost regular hypergraphs Hi =

(

[n − 1], {Ei1, . . . , E

iai})

, i = 1 . . . t. Recallthat the Ei

j are k-subsets of [n − 1].

Every vertex occurs in exactly m of the edges Eij and because Hi is almost regular

and m ≤ t it follows that every vertex of (Hi has degree 1 or 0 in Hi for all i, andthe edges {Ei

1, . . . , Eiai} are disjoint for all i.

The vertex {n − 1} ∈ [n − 1] occurs in exactly m of the edges Eij and because

the edges of Hi are disjoint for all i we can assume without loss of generality that{n − 1} ∈ Ei

1 for 1 ≤ i ≤ m. Define Di = Ei1\{n − 1}. Hence |Di| = k − 1 and

{Di}mi=1 =

([n−2]k−1

)

.

Let us first look at the case where m = t. Then {n − 1} ∈ Ei1 for all i. It follows

from (3.5) and (3.6) that q = s and ai ≥ 3 for all i. Define

Ri = {Ei2, . . . , E

iai}, (3.7)

then

{Ri}ti=1 =

(

[n − 1]

k

)

((

[n − 2]

k − 1

)

⊕ {n − 1}

)

=

(

[n − 2]

k

)

.

Notice that the set of edges of {Ri} is equal to the set of vertices V0.

We assumed that n > 3k whence also n−2 ≥ 3(k−1). By the induction hypothesiswe know that there exists a Hamilton cycle C1 through the Kneser graph K(n −2, k−1). Notice that the vertices of K(n−2, k−1) are equal to {Di, i = 1, . . . ,m}.Without loss of generality we can assume that

C1 = (D1,D2, . . . ,Dm), and

C2 =(

D1 ∪ {n − 1}, R11, . . . , R1(s−1), D1 ∪ {n}, D2 ∪ {n − 1}, . . . ,

Dm ∪ {n − 1}, Rm1, . . . , Rm(s−1), Dm ∪ {n})

,

is a cycle through all the vertices of V0 ∪ V1 ∪ V2 in K(n, k), where all the verticesRij, j = 1, ..., c of Ri are passed in arbitrary order.

34

Page 37: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

3.3. PROOF OF CONJECTURE 3.3 FOR THE CASE N > 3K

R11 U1 ∪ {n−1,n} R12 R1c ∪D1 {n}

D 2 ∪ {n−1}

D m

D1 ∪ {n−1}

Dm∪ {n}∪ {n−1}

Figure 3.4: The cycle C3: Note that Rij ∩ Rik = ∅ and that Ui ⊂ Di.

Next, we only have to add the vertices of V3 =([n−2]

k−2

)

⊕ {n − 1, n} to ob-tain a Hamilton cycle through K(n, k).

Define a bipartite graph B with partite sets U =([n−2]

k−2

)

and W =([n−2]

k−1

)

respec-tively. Let a vertex U ∈ U be adjacent to a vertex W ∈ W if and only if U ⊂ W .It follows that

dB(U) = (n − 2) − (k − 2) > k − 1 = dB(W ).

Assume there exists S ⊆ U with |S| > |N(S)|. Then

|S|(n − k) > |N(S)|dB(W ) = dB(N(S)) ≥ dB(S) = |S|(n − k),

a contradiction, and so by the theorem of Hall, there exists a matching M in B ofsize U . Hence for all U ∈ U there exists a Di = W ∈ W such that {U,W} ∈ Mfor some i. Then the following cycle is a Hamilton cycle in K(n, k):

C3 =(

D1 ∪ {n − 1}, R11, U1 ∪ {n − 1, n}, R12, . . . , R1c, D1 ∪ {n},

D2 ∪ {n − 1}, R21, U2 ∪ {n − 1, n}, R22, . . . , R2c, D2 ∪ {n}, . . . ,

Dm ∪ {n − 1}, Rm1, Um ∪ {n − 1, n}, Rm2, . . . , Rmc, Dm ∪ {n})

,

where Ui is skipped if it doesn’t exists (|U| < m). Notice that it is important that|Ri| ≥ 2 for all i, as we have defined in (3.6) and (3.7). In Figure 3.4 the cycle C3

is shown.

Now we only have to consider the case where m < t. Let

Ri =

{

{Ei2, . . . , E

iai} for 1 ≤ i ≤ m

{Ei1, . . . , E

iai} for m + 1 ≤ i ≤ t.

For a cycle through all the vertices of K(n, k) we have to add those Ri not inC3. That is, those Ri with i > m. We may assume that the element n − 1 is notcovered in the hypergraph Ht (otherwise we would have taken another element).

35

Page 38: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 3. KNESER GRAPHS

Dxj’Ey

Ry

Eyj^Ex

n−1 j n−1n

RxDxn

xU

Figure 3.5: The construction of a Hamilton cycle through K(n, k): Note that |Ri| ≥ 2,Dx ⊂ Ey

j , and zx /∈ Ex

j.

We will consider the special case where q = 1, t − 1 = m and s = 3 later on andignore it for now. Then it still holds that |Ri| ≥ 2.

Define a bipartite graph B with partite sets X = [m] and Y = [t]\[m] respectively.Let a vertex x ∈ X be adjacent to a vertex y ∈ Y if and only if Dx ⊂ Ey

j , forsome Ey

j ∈ Ry. Note that Dx is a (k − 1)-subset of [n − 2] and Eyj is a k-subset

of [n− 1] missing the vertex {n− 1}, whence each k-subset Eyj contains k distinct

(k − 1)-subsets Dx. It follows that

dB(y) = ay · k ≥

sk for y ≤ t − 2(s − 1)k if y = t − 12k if y = t.

dB(x) ≤ (n − 2) − (k − 1) = (s − 1)k + r ≤ sk − 1.

Assume there exists S ⊆ Y with |S| > |NB(S)|. Then

(|S| − 1)(sk − 1) > |NB(S)|maxx∈NB

(S)dB(x)

≥∑

y∈S

dB(y) ≥ sk(|S| − 2) + k(s − 1) + 2k.

This implies that |S| ≤ 1 − k, which is not possible, and by the theorem of Hall,there exists a matching M in B of size Y . Hence for all y ∈ Y there exists a differentx ∈ X such that {x, y} ∈ M . In other words, for all y ∈ [t]\[m] there exists x ∈ [m]such that Dx ⊂ Ey

j ∈ Ry for some j ≤ ay. Then because the number of edges ofRy is greater or equal than two and {n − 1} /∈ Ry, we know that there exists a j′

such that Eyj′ ∩ (Dx ∪ {n − 1}) = ∅. Furthermore let Ey

j \Dx = {zi} ∈ [n − 2]. We

know that there exists a j such that zi /∈ Exj. Now replace

Dx ∪ {n − 1}, Rx1, Ux ∪ {n − 1, n}, Rx2, . . . , Rxc, Dx ∪ {n}

in C3 by

Dx ∪ {n − 1}, Eyj′ , Ry, Ey

j , Exj, Ux ∪ {n − 1, n}, Rx, Dx ∪ {n},

then you’ve got a Hamilton cycle through K(n, k). See Figure 3.5.

Now we only have to consider the case where q = 1, t − 1 = m and s = 3. We

36

Page 39: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

3.4. THE CHVATAL-ERDOS CONDITION

already saw that if m < t then r ≥ 1. The sketched situation can never happenbecause in that case would n − 1 = 3k + r and

n − 1

k

(

n − 2

k − 1

)

=

(

n − 1

k

)

= 3

(

n − 2

k − 1

)

+ 1.

Combining these two gives (3k + r)m = (3m + 1)k resulting in rm = k which cannever happen for k > 1.

When n < 3k, the Kneser graph K(n, k) has no triangles. Chen [10] used Baranyai’spartition theorem again and found that all Kneser graphs are Hamiltonian for

n ≥1

2(3k + 1 +

5k2 − 2k + 1)

which is smaller than 2.62k+1.

3.4 The Chvatal-Erdos condition

As we have seen in Theorem 2.7, a condition for a graph G to be Hamiltonian isthat the independency number α(G) does not exceed the connectivity κ(G). Theindependency number of the Kneser graph K(n, k) is equal to

α(K(n, k)) =

(

n − 1

k − 1

)

.

To see this think of α as the size of a set A containing all k-subsets of n with onespecific element a ∈ {1, . . . , n}. You get

(

n−1k−1

)

subsets which aren’t adjacent forall pairs. This is the size of the independent set.We will see in section 3.6 that κ(K(n, k)) =

(

n−kk

)

. So now we have: If

(

n − 1

k − 1

)

(

n − k

k

)

,

then K(n, k) is Hamiltonian.

Chen and Lih [12] defined the following:

a(n, k) =

(

n−1k−1

)

(

n−kk

) and N0(k) = min{n | a(n, k) ≤ 1}.

Proposition 3.10. If a(n, k) ≤ 1 then a(n + 1, k) ≤ 1.

Proof. Let a(n, k) ≤ 1. Then

a(n + 1, k) =a(n, k)n(n + 1 − 2k)

(n − k + 1)2≤

n(n + 1 − 2k)

(n − k + 1)2≤ 1

37

Page 40: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 3. KNESER GRAPHS

k K(n, k) N0(k) n = ak + b k K(n, k) N0(k) n = ak + b

1 K(n, 1) 3 3k 11 K(n, 11) 75 6k+92 K(n, 2) 6 3k 12 K(n, 12) 86 7k+23 K(n, 3) 11 3k+2 13 K(n, 13) 97 7k+64 K(n, 4) 16 4k 14 K(n, 14) 110 7k+125 K(n, 5) 22 4k+2 15 K(n, 15) 123 8k+36 K(n, 6) 29 4k+5 16 K(n, 16) 136 8k+87 K(n, 7) 37 5k+2 17 K(n, 17) 150 8k+148 K(n, 8) 45 5k+5 18 K(n, 18) 165 9k+39 K(n, 9) 55 6k+1 19 K(n, 19) 180 9k+910 K(n, 10) 64 6k+4 20 K(n, 20) 196 9k+16

Table 3.1: Hamiltonian Kneser Graphs

Now we’ve got the following theorem:

Theorem 3.11. For fixed k the graph K(n, k) is Hamiltonian if n ≥ N0(k).

The results for N0(k) are shown in Table 3.1 for k = 1, . . . , 20. Unfortunatelythere are no results here for n < 3k. Note that N0(k) always exists because forsufficiently large n

(

n − 1

k − 1

)

∼ nk−1 and

(

n − k

k

)

∼ nk.

3.5 The Odd Graph

Definition 3.12. The Odd graph Ok+1 is the graph K(2k + 1, k)

The Odd graph is the Kneser graph K(n, k) where n is minimal under the conditionthat the graph is connected. There has been done some research to Odd graphsbut we are nowhere near a proof of the conjecture that almost all Odd graphs haveHamilton cycles (Note that O2 is equal to the Petersen graph). The proofs wegave so far for Hamiltonian Kneser graphs do not concern Odd graphs, except forthe triangle K(3, 1). So far the Hamiltonicity of the Odd graph has been provenfor k starting at 3 up to 7: Balaban [5] exhibited Hamilton cycles for k = 3 andk = 4, Meredith and Lloyd [32] established Hamilton cycles for k = 5 and k = 6and Mather [31] showed a Hamilton cycle for k = 7. These proofs are basicallyconstructions of Hamilton cycles and does not provide a method for proving thatall Odd graphs have Hamilton cycles. We do have the much stronger conjectureof Meredith and Lloyd [33] about k-ply Hamiltonian Odd graphs.

Definition 3.13. A graph is called m-ply Hamiltonian if it contains m-edge dis-joint Hamilton cycles.

38

Page 41: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

3.6. THE UNIFORM SUBSET GRAPH

Conjecture 3.14 (Meredith and Lloyd). The graphs O2k and O2k+1 are k-plyHamiltonian for all k ≥ 2.

3.6 The Uniform Subset Graph

Definition 3.15. The uniform subset graph G(n, k, t) has as vertices the k-subsetsof {1, 2, . . . , n} where two vertices are adjacent if the k-subsets have t elements incommon.

Definition 3.16. A graph G = (V,E) is edge transitive if for all e, f ∈ E, thereexists an automorphism of G that maps the endpoints of e to the endpoints of f .

Theorem 3.17 (Chen and Lih [12]). G(n, k, t) is edge transitive.

Proof. Let {A,B} be an edge of G(n, k, t). Note that the sets A ∩ B, A\B andB\A are pairwise disjoint. Let {C,D} be another edge of G(n, k, t). There exists apermutation f of {1, . . . , n} that maps A∩B onto C∩D, A\B onto C\D and B\Aonto D\C. Hence, f(A) = C, f(B) = D and {f(A), f(B)} ∈ E and G(n, k, t) isedge transitive.

Proposition 3.18. The connectivity κ of G(n, k, t) is equal to(

kt

)(

n−kk−t

)

, wheret < k.

We know that the degree of a vertex v in G(n, k, t) is equal to

dG(v) =

(

k

t

)(

n − k

k − t

)

,

so we want to prove that κ(G) = dG(v). Chen and Lih [12] prove this is true byuse of a theorem of Lovasz [30] that states that all connected graphs with an edgetransitive automorphism group and with all degrees at least r, is r-connected.

3.6.1 The Johnson graph

Definition 3.19. The Johnson graph J(n, k) is the graph G(n, k, k − 1).

The Johnson graph is derived from the Johnson distance, which is the number ofelements between two k-subsets, the k-subsets differ.

Theorem 3.20. All Johnson graphs are Hamiltonian.

Notice that finding a Hamilton cycle in a Johnson graph is the same as finding aGray code as we did in Section 3.2. So we already proved the theorem above. Inthe following, we want to give another proof as well and for this let us first provethe following induction theorem of Chen and Lih [12]:

39

Page 42: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 3. KNESER GRAPHS

B A

BA

x

y

a

b

C1C2∪

a0 a1

a0 b1b1n+1

a1n+1

+1}n{

Figure 3.6: The Hamilton cycle through G(n + 1, k + 1, t + 1)

Theorem 3.21. If both G(n, k, t) and G(n, k + 1, t + 1) are Hamiltonian, so isG(n + 1, k + 1, t + 1)

Proof. Assume G(n, k, t) and G(n, k+1, t+1) are Hamiltonian. Hence there existsa cycle C1 through all the vertices of G(n, k, t) and there exists a cycle C2 throughall the vertices of G(n, k + 1, t + 1). Notice that if you add the element {n + 1} toall the k-subsets of n, then C1 is isomorphic to a cycle of length

(

nk

)

through all thevertices containing the element {n + 1} in G(n + 1, k + 1, t + 1). Furthermore, C2

is a cycle of length(

nk+1

)

in G(n + 1, k + 1, t + 1) through all the vertices withoutthe element {n + 1}.

Let {a, b} be an edge of C2 and let a0 ∈ a ∩ b, a1 ∈ a\b and b1 ∈ b\a. Considerthe vertices

x = {n + 1, a1} ∪ b\{a0, b1} and y = {n + 1, b1} ∪ a\{a0, a1}.

Then x and y are connected and because the uniform subset graph is edge tran-sitive, we can assume that the edge {x, y} is subject of the cycle C1. Then thecycle

C1\{x, y} ∪ {x, a} ∪ C2\{a, b} ∪ {b, y}

is a Hamilton cycle (of length(

nk

)

+(

nk+1

)

=(

n+1k+1

)

) in G(n + 1, k + 1, t + 1), seeFigure 3.6.

Theorem 3.22. Let k be fixed and

n0 = min{n| G(m,k, 0) is Hamiltonian ∀m ≥ n}.

If G(n0, k + r, r) is Hamiltonian for r = 0, . . . , n0 − 2k − 1, then G(n, k + r, r) isHamiltonian for r = 0, . . . , n − 2k and n ≥ n0.

40

Page 43: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

3.6. THE UNIFORM SUBSET GRAPH

Proof. Fix k and n0 such that G(n, k, 0) is Hamiltonian for all n ≥ n0. Noticethat n0 exists because of the results on Kneser graphs (i.e. Theorem 3.9). AssumeG(n0, k + r, r) is Hamiltonian for r = 0, . . . , n0 − 2k − 1. Notice that the graphsG(n, k, t) and G(n, n−k, n−2k + t) are isomorphic because of the complementarysubsets. So we know that G(n, n−k, n−2k) is Hamiltonian for all n ≥ n0. It followsdirectly by Theorem 3.21 that G(n, k + r, r) is Hamiltonian for r = 0, . . . , n − 2k.For picturing this, see Figure 3.7.

(n0, k, 0) (n0, k + 1, 1) . . . . . . (n0, n0 − k, n0 − 2k)(n0 + 1, k, 0) (n0 + 1, k + 1, 1) . . . . . . (n0 + 1, n0 − k, n0 − 2k) . . .

......

......

......

(n, k, 0) (n, k + 1, 1) . . . . . . (n, n0 − k, n0 − 2k) . . .

. . . (n0 + 1, n0 + 1 − k, n0 + 1 − 2k)...

. . ....

. . .

. . . (n, n0 + 1 − k, n0 + 1 − 2k) . . . . . . (n, n − k, n − 2k)

Figure 3.7: G(n, k + r, r) is Hamiltonian for all n ≥ n0 and r = 0, . . . n − 2k.

Proof of Theorem 3.20. We want to prove that G(n, k, k − 1) is Hamiltonian forall admissible (n, k, k − 1). We know that G(n, 1, 0) is Hamiltonian for all n ≥ 3.Furthermore, G(3, 2, 1) is a triangle and obviously Hamiltonian. So by Theorem3.22 we now know that G(n, 1+r, r) is Hamiltonian for r = 0, 1, . . . , n−2. In otherwords, G(n, k, k − 1) is Hamiltonian for all admissible (n, k, k − 1) and n ≥ 3.

3.6.2 Other Hamiltonian Subset Graphs

Let N1(k)= min{n|(

nk

)

≤ 3k(

n−kk−1

)

}. Chen and Lih [12] show by direct computa-

tions that(

nk

)

≤ 3k(

n−kk−1

)

when n = k2 − k and k ≥ 3, so N1(k) exists for k ≥ 3.By Theorem 2.4 we now have the following theorem:

Theorem 3.23. The graph G(n, k, 1) is Hamiltonian for all n ≥ N1(k).

Proof. First note that for k = 2 it easily follows from Theorem 3.21 that G(n, 2, 1)is Hamiltonian for n ≥ 3. Let k ≥ 3, n ≥ 2k − 1 and N1(k)= min{n|

(

nk

)

3k(

n−kk−1

)

}. The results for N1(k) are shown in Table 3.2 for k = 2, . . . , 20. We

41

Page 44: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 3. KNESER GRAPHS

k G(n,k,1) N1(k) n=ak+b k G(n,k,1) N1(k) n=ak+b

11 G(n, 11, 1) 74 6k+82 G(n, 2, 1) 3 k+1 12 G(n, 12, 1) 88 7k+43 G(n, 3, 1) 6 2k 13 G(n, 13, 1) 104 8k4 G(n, 4, 1) 10 2k+2 14 G(n, 14, 1) 121 8k+95 G(n, 5, 1) 15 3k 15 G(n, 15, 1) 139 9k+46 G(n, 6, 1) 22 3k+4 16 G(n, 16, 1) 159 9k+157 G(n, 7, 1) 29 4k+1 17 G(n, 17, 1) 180 10k+108 G(n, 8, 1) 39 4k+7 18 G(n, 18, 1) 202 11k+49 G(n, 9, 1) 49 5k+4 19 G(n, 19, 1) 225 11k+1610 G(n, 10, 1) 61 6k+1 20 G(n, 20, 1) 250 12k+10

Table 3.2: Hamiltonian Subset Graphs

see that N1(k) increases faster than k and N1(k) ≤ (k − 1)k + 1 ≤ k2 for allk. If n ≤ k2, some regular calculations show us that if n ≥ N1(k), G(n, k, 1) isHamiltonian.

If we look at the results in Table 3.1, we see that N0(k) increases faster than k aswell and N0(k) ≤ k2 + 2 for all k. So we have N0(k − 1) ≤ (k − 1)2 + 2 ≤ k2 andit follows from Section 3.4 that G(k2, k − 1, 0) is Hamiltonian. Together with theabove, Theorem 3.21 tells us that G(n, k, 1) is Hamiltonian for all n ≥ k2.

Hence now we have that G(n, k, 1) is Hamiltonian for all n ≥ N1(k) and thetheorem is proved.

If we make this method more general we have the following:Let k ≥ t + 1 and Nt(k)= min{n|

(

nk

)

≤ 3(

kt

)(

n−kk−t

)

} and let ft(k) be an upper

bound for Nt(k). If Nt(k) exists, ft(k) ≤ k2/t and ft−1(k) ≤ k2/t then G(n, k, t) isHamiltonian for n ≥ Nt(k). Unfortunately we do not have an expression for ft(k)so along this way, there is no easy way to check the Hamiltonicity for G(n, k, t).

If we go back to Theorem 3.22 we also see that:

i) G(n, 2, 0) is Hamiltonian for all n ≥ 6. Furthermore, G(6, 3, 1) is Hamiltoni-anand G(6, 4, 2) is not connected so no admissible pair. Hence G(n, 2 + r, r) isHamiltonian for r = 0, 1, . . . , n − 4 and n ≥ 6. In other words, G(n, k, k − 2) isHamiltonian for all admissible (n, k, k − 2) with n ≥ 6.

ii) G(n, 3, 0) is Hamiltonian for all n ≥ 7, see Heinrich and Wallis [18]. Fur-thermore, G(7, 3, 0) and G(7, 4, 1) are Hamiltonian (in fact, they are isomorphic).Hence G(n, 3 + r, r) is Hamiltonian for r = 0, 1, . . . , n − 6 and n ≥ 7. In otherwords, G(n, k, k − 3) is Hamiltonian for all admissible (n, k, k − 3) with n ≥ 7.

42

Page 45: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

Chapter 4

Cayley graphs

The Cayley graph is named after Arthur Cayley (1821-1895) and occurs todayfrequently as a model of interconnection networks.

Definition 4.1. Let G be a finite group and S be a minimal generating set of G.A Cayley graph Cay(G,S) is a graph with as vertices the elements g ∈ G. Twovertices g1 and g2 are adjacent if g2 = g1 · s, where s ∈ S

Note that a generating set S is minimal if S generates G but no proper subset ofS does.

Theorem 4.2 (Gosil and Royle, [17]). All Cayley graphs are vertex transitive.

Proof. Let G be a group and X = (V,E) be the Cayley graph of some set ofgenerators S of G. So V = G and {g1, g2} ∈ E if g2 = g1 · s, where s ∈ S. Let fbe the automorphism of G given by

f(g) := xg for some x ∈ G.

Let {g, h} ∈ E. We are going to prove that {f(g), f(h)} ∈ E. Without loss ofgenerality we can assume that h = gs for some s ∈ S. It follows that

f(h) = xh = xgs = f(g)s,

and {f(g), f(h)} ∈ E under f and f is an automorphism.

If you want to particularly map some element g to g′, take x = g′ · g−1 whichmakes each Cayley graph vertex transitive.

Ignoring K2, the known non-Hamiltonian vertex transitive graphs are not Cayleygraphs. This leads to the following conjecture:

Conjecture 4.3. All Cayley graphs are Hamiltonian.

43

Page 46: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 4. CAYLEY GRAPHS

Figure 4.1: C3, C5 and C3 × C5

The advantage of this conjecture above Conjecture 1.2 is that Cayley graphs cor-respond to a finite group G and a generating set S. Thus one can ask for whichG and S the conjecture holds rather than attack it in full generality.In this chapter we prove the Hamiltonicity of all Cayley graphs based on anyabelian group. We also see some results of Cayley graphs of a dihedral or sym-metric group.

4.1 Abelian groups

As you will expect, we start to look at the Cayley graphs based on some set ofgenerators of an abelian group. Proofs that these graphs have Hamilton paths canbe found in [41], [42], and [21].

4.1.1 Cycle graphs

The cycle graph Cn is a graph on n vertices containing a single cycle throughall vertices. All Cayley graphs based on a generating set of a single element areisomorphic to a cycle graph. Groups that can be generated by a single element(the cyclic groups) are of course abelian.

The Cartesian Product Cn1 × · · · ×Cnkof k cycles can be seen as an n1 × · · · ×nk

grid where the vertex (i1, . . . , ik) is adjacent only to the vertices

(i1 ± 1 mod n1, i2, . . . , ik), . . . , (i1, i2, . . . , ik ± 1 mod nk).

Proposition 4.4. The cartesian product Cn1×· · ·×Cnkof k cycles is Hamiltonian.

Proof. Proof by induction to k. If k = 1 the lemma is trivial. Assume the lemmaholds for k = m. Let X = Cn1 × · · · × Cnm+1 and assume first at least onea ∈ {n1, . . . , nm+1} is even. Without loss of generality we can assume nm+1 is even.Let P = (v1, . . . , vl) with v1 = vl, the Hamilton cycle through Cn1 × · · · × Cnm.Then

P ′ = ((v1, 0), . . . , (vl−1, 0), (vl−1, 1), . . . , (v1, 1), . . . , (v1, nm+1 − 1))

is a Hamilton cycle through X, see Figure 4.2.

44

Page 47: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

4.1. ABELIAN GROUPS

Figure 4.2: Hamilton cycles through Cn1× · · · × Cnk

.

Let now all a ∈ {n1, . . . , nm+1} be odd and again let P = (v1, . . . , vl) with v1 = vl,be the Hamilton cycle through Cn1 × · · · × Cnm . Then

P ′ =((v1, 0), . . . , (v1, nm+1 − 1), (v2, nm+1 − 1), . . . , (v2, 1), . . . , (vl−2, 1),

. . . , (vl−2, nm+1 − 1), (vl−1, nm+1 − 1), (vl−1, 0), . . . , (v2, 0))

is a Hamilton cycle through X, see Figure 4.2.

We can conclude that if X is a Cayley graph and X ∼=∏k

i=1 Ck then X is Hamil-tonian. Thus, if G is a group equal to the product of cyclic groups, then G has atleast one Hamiltonian Cayley graph.

4.1.2 General abelian groups

It leaves us to prove that all Cayley graphs of all abelian groups are Hamiltonian.We first give an idea of the proof.

Let G be an abelian group and S a minimal set of generators. If S consists of oneelement, the group G is cyclic and thus Hamiltonian. Every new generator givesus cycles in the Cayley graph; the cycles are of length of the order of the generator.We can ’build’ a grid in the same way as we did before by using the generators ascoordinates. We stop in each direction if we meet an element we met before byuse of the preceding generators.

Theorem 4.5. Let G be an abelian group and S = {s1, ..., sr} a minimal set ofgenerators. Let, for i = 2, . . . , r, vi be the smallest integer such that

visi =

i−1∑

j=1

ajsj

for some numbers aj , and let v1 be equal to the order of s1. Then the Hamiltoncycle in the grid

A = Cv1 × Cv2 × · · · × Cvr (4.1)

generates a Hamilton cycle in Cay(G,S) as well.

45

Page 48: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 4. CAYLEY GRAPHS

Proof. It is not hard to see that if vi is equal to the order of si for all i, thenCay(G,S) is isomorphic to the product of cycles Cvi

with v1v2 · · · vr = n. If this isnot the case we still want to prove that every element of Cay(G,S) is contained inthe grid (4.1) exactly once. We prove this by induction to the number of generators.It is immediate that if r = 1 no element x ∈ G is contained in A more than once.Assume for r = k − 1 no element is contained in A more than once. Let r = kand suppose there is an element x of G that is contained in the grid A more thanonce. Then there are ci, di < vi such that

x =k

i=1

cisi =k

i=1

disi,

where at least for some i holds that ci 6= di. Recall that vk is the minimal integersuch that

vksk =k−1∑

i=1

aisi.

We consider three different cases:

i) If ck = dk, then

y = x − cksk = x − dksk,

and y is an element that is contained in Cv1 × Cv2 × · · · × Cvk−1more than once,

contradicting our induction hypothesis.

ii) If ck < dk, then

(dk − ck)sk =

k−1∑

i=1

(ci − di)si,

but then dk − ck < dk < vk contradicting the fact that vk was minimal.

iii) If ck > dk the proof goes the same as in ii).

Hence we have proved that no element of G is contained in the grid A more thanonce. It remains to prove that every element of G is contained in the grid A atleast once. Suppose there is an element x ∈ G which does not lay in our grid. So

x = a11s1 + a12s2 + . . . + a1rsr

with a1i ≥ vi for at least one i. Let j be the largest index for which a1j ≥ vj, thenwe can write

x = a11s1 + . . . + vjsj + a2jsj + . . . + a1rsr

= a11s1 + . . . +

j−1∑

i=1

a2isi + a2jsj + . . . + a1rsr

= (a11 + a21)s1 + . . . + (a1(j−1)a2(j−1))sj−1 + a2jsj + . . . + a1rsr.

46

Page 49: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

4.2. THE PRODUCT OF HAMILTONIAN CAYLEY GRAPHS

Figure 4.3: Z6 and Z12 both with generators 2 and 3

We can continue this process until all coefficients bi = a1i + . . . + aki are smallerthan vi, contradicting our assumption that x lay outside our grid A.

A final note must be made by the grid we use. In the previous section we sawthe Hamilton cycle through such a grid, with the only difference that for eachCvi

, the vertex vi − 1 is connected to the vertex 0. With the ’cutting of’ processwe used in this proof, this is not longer the case except at least for the firstgenerator. Fortunately, if we take a look at the Hamilton cycles in Figure 4.2, thisis enough.

We can conclude that every Cayley graph of any abelian group is Hamiltonian.

An example of a Cayley graph based on an abelian group is the Circulant graph.The Circulant graph Circ(n, s1, ..., sr) is a graph on n vertices in which the i-thvertex is adjacent to the (i+ j-)th and (i− j)-th vertices for each j in a certain listL = {s1, ..., sr}. The Circulant graph on n vertices is equal to the Cayley graphbased on the group Zn with generators s1, ..., sr. Theorem 4.5 gives us a Hamiltoncycle through Circ(n, s1, ..., sr).

4.2 The product of Hamiltonian Cayley graphs

We now take a look at the product of Cayley graphs. So far, we only saw thecartesian product of graphs: In Section 4.1.1 we saw that if X en Y are graphswith vertices xi ∈ X, yi ∈ Y , then two vertices (x1, y1), (x2, y2) ∈ X × Y areadjacent if x1 is adjacent to x2 in X or if y1 is adjacent to y2 in Y . If we want tosay something about some Cayley graph of the product of groups G×H, we needanother definition of products of graphs.

Definition 4.6. The conjunction product Cay(G,S) · Cay(H,T ) of two Cayleygraphs is the graph Cay(G × H,S × T ).

We see that if X en Y are graphs with vertices xi ∈ X, yi ∈ Y , then two vertices(x1, y1), (x2, y2) ∈ X · Y are adjacent if x1 is adjacent to x2 in X and if y1 isadjacent to y2 in Y .

47

Page 50: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 4. CAYLEY GRAPHS

����

����

��������

����

��

��������

��

����

1i

j

k

−1

−i

−j

−k

Figure 4.4: The Cayley graph of the quaternion group with generating set {i, j}. All

Cayley graphs of the quaternion group are isomorph and it is easy to find a Hamilton

cycle.

Conjecture 4.7 (Keating, [25]). If there is a Hamilton cycle in each of Cay(G,S)and Cay(H,T ), then there is a Hamilton cycle in the conjunction

Cay(G,S) · Cay(H,T ),

unless it is not connected.

Keating proved this conjecture is true for all cases except the case where Cay(G,S) =C2, which is still open. We also see that if Cay(G,S) = C2, H is of odd order, and(v0, . . . , vk = v0) is a Hamilton cycle through Cay(H,T ), then

((0, v0), (1, v1), (0, v2), . . . , (1, vk), (0, v1), . . . , (0, vk))

is a Hamilton cycle through Cay(G,S)·Cay(H,T ). Hence, the only open case ofConjecture 4.7 is the case where Cay(G,S) = C2 and H is of even order.

An example of a Cayley graph based on the product of groups is the Cayley graphof a Hamiltonian group. A Hamiltonian group is a non-abelian group for whichevery subgroup is normal. Every Hamiltonian group G is a direct product of theform

G = Q8 × B × A,

where Q8 is the quaternion group, B is a number of copies of Z2 and A is an oddorder abelian group (Alspach and Qin [1]). It is easy to show that Cay(Q8, S) isHamiltonian for all generating sets S (see Figure 4.2). Furthermore, we know thatB × A is an abelian group and we proved in the previous section that all Cayleygraphs of this group are Hamiltonian. Then, by the results of Conjecture 4.7, allCayley graphs of Q8 × (B × A) are also Hamiltonian.

4.3 The Dihedral group

The dihedral group Dn is the group of symmetries of a regular polygon with nsides, including both rotations and reflections. A regular polygon with n sides has

48

Page 51: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

4.3. THE DIHEDRAL GROUP

2n different symmetries: n rotational symmetries and n reflection symmetries. Ingeneral, Dn has elements

R0, ..., Rn−1 and S0, ..., Sn−1,

with I = R0 = Rn, I = S2i for all i and SiRjSi = R−j for all i, j. Furthermore,

composition is given by the following:

RiRj = Ri+j, RiSj = Si+j, SiRj = Si−j, SiSj = Ri−j.

It should be clear that the dihedral group is not abelian and a generating setshould contain at least one reflection symmetry.

Proposition 4.8. If a set A = {a1, . . . , ak} is a minimal generating set of Zn,then

A = {Ra1 , Ra2 , . . . , Rak, Sak

} (4.2)

is a minimal generating set of Dn.

Proof. Let A be a minimal generating set of Zn. Then every element g ∈ Zn canbe written as g = n1a1 + . . . + nkak for some numbers ni. It follows that for everyRg, Sg ∈ Dn,

Rg = Rn1a1+...+nkak= Rn1a1 · · ·Rnkak

= Rn1a1

· · ·Rnkak

,

Sg = Sn1a1+...+nkak= Rn1a1 · · ·Rnk−1ak−1

· Snkak= Rn1

a1· · ·R

nk−1ak−1 Sak,

and A is a generating set. Every rotation symmetry Rg can be written as acombination of the rotation symmetries of A because

SakRmiai

Sak= R−miai

= R−miai

.

To prove that A is minimal, we suppose that it is not. Then there exists i ∈{1, . . . , k} and there are mj with

Rai= Rm1a1 · · ·Rmi−1ai−1 · Rmi+1ai+1 · · ·Rmkak

= Rm1a1+...+mi−1ai−1+mi+1ai+1+...+mkak

and ai = m1a1 + . . . + mi−1ai−1 + mi+1ai+1 + . . . + mkak, contradicting the factthat A is minimal.

It follows directly from the fact that all Cayley graphs Cay(G,S) with G abelianare Hamiltonian, that if A is of the form (4.2) then Cay(G,S) is Hamiltonian.

Theorem 4.9. Let Cay(Dn, A) be a Cayley graph of a dihedral group Dn whereA contains at least one rotation. Then Cay(Dn, A) is Hamiltonian.

49

Page 52: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 4. CAYLEY GRAPHS

Proof. Let A be a minimal generating set of the dihedral group Dn containing atleast one rotation. Let G be the subgraph of the Cay(Dn, A) with vertex set Dn

and edge setA′ = {{x, y}| yx−1 = Ra, x, y ∈ Dn, Ra ∈ A}.

Then we know that G consists of components G1, . . . , Gm, which are all isomorphicto a circulant graph Zk. Note that k is equal to the number of vertices generated bythe rotations in A and m = 2n/k. We saw in section 4.1.2 that these componentscontain a cycle Ck. Let {Sa1 , . . . , Sal

} be the reflections in A and let

G = G ∪ {{u, v}| vu−1 = Sa1 , u, v ∈ Dn}.

Let a vertex u1 in a component Gp be joined with a vertex v1 in a component Gq.Then there is for all x ∈ Gp a vertex y ∈ Gq with Sa1x = y. More specifically,because Sa1RaSa1 = R−1

a , it follows that if u′1 is a neighbor of u1 in Gp, then

Sa1u′1 = v′1 and v′1 is a neighbor of v1. Hence by adding edges generated by Sa1 to

G, we get m/2 components with each components containing a cycle C2k.

Go on by adding the edges generated by Sa2 . Let {u2, u′2} be an edge of a compo-

nent Gp of G other than {u1, u′1} with u′

2u−12 = Ra for some Ra ∈ A. We again see

that u2 and u′2 have neighbors v2 = Sa2u2 and v′2 = Sa2u

′2 respectively, in another

component Gq with v′2v−12 = R−1

a .We can go on with this procedure by adding edges generated by Sa3 , . . . , Sal

oneat a time. Start each time with an edge generated by a rotation symmetry. Af-ter adding the edges of each reflection, the components become larger but remaincontain cycles through the entire component. We need all reflections for makingthe graph connected because A is minimal and we end up with a cycle through allvertices of Dn.

Hence now we know that all Cayley graphs of a dihedral group are Hamiltonian ifthe generating set contains at least one rotation. Our further considerations canbe restricted to the case that the generating set contains only reflections.

If S = {Si, Sj} is a generating set for Dn, then

C = (I, Si, SjSi, . . . , Si(SjSi)n−1, (SjSi)

n = I) (4.3)

is a Hamilton cycle and Cay(Dn, S) is Hamiltonian for all generating sets contain-ing two elements. As a result we have

Proposition 4.10 (Holsztynski and Strube [21]). All Cayley graphs of the dihedralgroup Dp, where p is a prime, are Hamiltonian.

Proof. Let p be a prime number. The only minimal generating sets of Dp areeither of the form A1 = {Ra1 , Sa2} or A2 = {Sa1 , Sa2}. We have already shownthat Cayley graphs of the dihedral group with these forms of generating sets areHamiltonian.

50

Page 53: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

4.4. THE SYMMETRIC GROUP

In fact, Witte [43] showed that every Cayley graph on a group of prime-powerorder greater than two is Hamiltonian.

Alspach and Zhang [2] have proved that Cay(Dn, S) is Hamiltonian for S ={Si, Sj, Sk} and hereby proved that all cubic Cayley graphs based on a dihedralgroup are Hamiltonian. Unfortunately there are no results known for the casewhere the generating set consist of more than three reflections. The problem lies,of course, in the non-abelian property of the reflection symmetries.

4.4 The Symmetric group

The symmetric group Sn is the group of all the permutations of {1, . . . , n}. Thegroup Sn has order n! and is not abelian for n > 2. We already saw Dn, whichis a subgroup of Sn. The symmetric group occurs in change ringing : Ringing alln bells in a bell tower in one of the n! possible ways is called a change. Ringingall n! different changes is called an extent. Change ringing is the art of ringingal n! possible changes such that no change differs in its order of ringing by morethan two positions from one change to the next. Two bells can only be swapped intheir orders and the last ringing change must be equal to the first. Change ringingemerged in England in the 17th century and is, particularly in England, still apopular occupation. The group theoretical underpinnings of change ringing havebeen pursued by mathematicians who have done a lot of study on change ringingthroughout the years since. White [40] has shown that the constraints give rise toa set of transpositions in Sn and that an extent exists if and only if the Cayleygraph on Sn generated by these transpositions is Hamiltonian.

Definition 4.11. An involution is a group element of order two.

Definition 4.12. A cycle is a permutation f for which there exists an element xin {1, . . . , n} such that x, f(x), . . . , fk(x) = x are the only elements moved by f . kis called the length of the cycle and is equal to its order. Cycles of length two arecalled transpositions.

Note that this definition of a cycle has nothing to do with a cycle in a graph andthat an involution need not always be a transposition.

Theorem 4.13 (Pak and Radoicic, [36]). If Sn is generated by three involutionsα, β, γ such that two of them commute, then the Cayley graph Cay(Sn, {α, β, γ})is Hamiltonian.

Proof. Let α, β, γ be three involutions that generate Sn and let αβ = βα. Define

dy(Xi) = {g ∈ Sn − Xi| g = xy, x ∈ Xi},

51

Page 54: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

CHAPTER 4. CAYLEY GRAPHS

Figure 4.5: A spanning cycle through Xi+1. Note that it is important that αβ = βα.

where X1 is equal to the dihedral group generated by β and γ,

Xi+1 = Xi ∪ yiDn and yi ∈ dα(Xi) ⊂ Sn − Xi.

We are going to prove by induction that there exists a spanning cycle of Xi for alli. The proof is based on the same idea as the proof we gave for Theorem 4.9.

We already saw in (4.3) a spanning cycle through X1. Note that, by definition ofa group, dβ(X1) = dγ(X1) = ∅ and at this point, new vertices can only be reachedby the use of α. Assume there exists a spanning cycle of Xi. It still holds thatdβ(Xi) = dγ(Xi) = ∅ and if

dα(Xi) = ∅ (4.4)

as well, then it follows that Xi = Sn an we have found a Hamilton cycle throughCay(Sn, {α, β, γ}). Assume dα(Xi) 6= ∅ and let yi ∈ dα(Xi). Then Xi ∩ yiDn = ∅,otherwise would yih = x ∈ Xi for some h ∈ Dn. This implies that yi ∈ Xi

which is a contradiction. Hence, only ’new’ vertices are added to Xi to obtainXi+1 = Xi ∪ yiDn. By the induction hypothesis, there exists a spanning cycle C1

through Xi and by (4.3) there exists a spanning cycle C2 through yiDn. We knowthat yiα = xα2 = x for some x ∈ Xi and because yi = xα /∈ Xi, x is on C1 andconnected to xβ and xγ. On the other hand yi is in Xi+1 so on C2 and connectedto yiβ and yiγ. Then because αβ = βα

(C1\{x, xβ} ∪ C2\{yi, yiβ})

is a spanning cycle through Xi+1. If we continue this process un till (4.4) holds,we will find our Hamilton cycle through Sn.

Another known result is the following:

Theorem 4.14 (Kompel’makher and Liskovets, [27]). The Cayley graph Cay(Sn, S)is Hamiltonian if S consists only of transpositions.

Furthermore, it is proved that G is Hamiltonian if S consists of either n−1 right ro-tations: {(12), (123), . . . , (12 · · · n)} or n− 1 left rotations: {(21), (321), . . . , (n n−1 · · · 1) (Jiang and Ruskey [23]).

52

Page 55: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

4.4. THE SYMMETRIC GROUP

A problem which remained open for a long time is the one where S containsone cycle of length n and one transposition. Compton and Williamson [15] wereable to construct a Hamilton cycle in this graph using Gray Codes, as we did forsome particular Kneser graphs.

Although the Hamiltonicity of Cay(Sn, S) is proved for many S, there are stillmany more open cases in order of this problem.

53

Page 56: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part
Page 57: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

Chapter 5

Prime graphs

There are lots of results known about the Hamiltonicity of vertex transitive graphswhere the order of the graph is some function of a prime number p. For instance,Alspach [3] proved that with exception of the Petersen graph, all connected ver-tex transitive graphs of order 2p are Hamiltonian. And Kutnar and Marusic [28]proved that with exception of the Coxeter graph, all connected vertex transitivegraphs of order 4p are Hamiltonian.

Furthermore, if we restrict ourself to Cayley graph, we have for instance the fol-lowing results:

Theorem 5.1 (Witte, [43]). Every Cayley graph on a group of prime-power ordergreater than 2, and any generating set is Hamiltonian.

Witte uses the following theorem about quotient groups:

Theorem 5.2. Let S be a generating set for a group G and N a normal subgroupof G. Let S be the image of S in the quotient group G/N . If there is a Hamiltoncycle C = (a1, a2, . . . , an) in Cay(G/N, S) such that a1a2 · · · an generates N , then|N | copies of C gives a Hamilton cycle in Cay(G,S).

The following theorem of Jungreis, [24] ensures the Hamiltonicity of many Cayleygraphs.

Theorem 5.3. Let p and q be prime numbers. Every Cayley graph on a group oforder pq, 4q (q > 3), p2q (2 < p < q), 2p2, 2pq, 8p or 4p2, and any generating setis Hamiltonian.

We already saw in Proposition 4.10 an example of such a theorem.

55

Page 58: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part
Page 59: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

Epilogue

In this thesis I studied some properties of vertex transitive graphs with the purposeto discuss the existence of Hamilton cycles in such graphs. In particular, I studiedthe structure of Kneser graphs in more detail and found out that the only n forwhich it is not yet proved that Kneser graphs K(n, k) are Hamiltonian is when2k +1 ≤ n ≤ 2, 62k +1. During this study, I proved Baranyai’s partition theorem,the existence of Gray codes and some results about uniform subset graphs.In the last part of my thesis, I did some research on Cayley graphs and the existenceof Hamilton cycles in such graphs. I started by proving that all Cayley graphs ofany abelian group have Hamilton cycles. From here on, I found out about theresults known of Hamiltonicity concerning the product of groups, Hamiltoniangroups, dihedral groups and symmetric groups.

When I started my master’s thesis, I, of course, had to pick a subject first. I alwaysloved graph theory and combinatorics so that choice was already made. There area lot of open problems in any kind of mathematics. Graph theory is no different.I chose for a problem with some algebraic aspects and the conjecture of Lovaszseemed a suitable problem.On the one side the problem appeared to be very broad and I had to decidewhich way I wanted to go in my research. On the other hand the problem wasalready fully investigated by lots of people and it seemed really hard to make acontribution. Nevertheless, I found my way to contribute to the problem; I workedout some results known and found some holes to close.

Thanks goes to my supervisor Dion, who is always online, and to my friend Bart,who always likes to think with me.

57

Page 60: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part
Page 61: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

Bibliography

[1] B. Alspach, Y. Qin, Hamilton Connected Cayley Graphs on HamiltonianGroups, European Journal of Combinatorics 22: 777-787 (2001)

[2] B. Alspach, C. Q. Zhang, Hamilton Cycles in cubic Cayley Graphs on DihedralGroups, Ars Cominatorics 28: 101-108 (1989)

[3] B. Alspach, Hamiltonian Cycles in Vertex Transitive Graphs of Order 2p,Proceedings of the Tenth Southeastern Conference on Combinatorics, GraphsTheory and Computing XXIII-XX: 131-139 (1979)

[4] W.N. Anderson, T.D. Morley, Eigenvalues of the Laplacian of a Graph, Linearand Multilinear Algebra 18: 141-145 (1985)

[5] A. T. Balaban, Chemical Graphs, Part XIII: Combinatorial Patters, Rev.Roumaine Mathematical Pures Application 17: 3-16 (1972)

[6] Z. Baranyai, The Edge-Coloring of Complete Hypergraphs I, Journal of Com-binatorial Theory B 26: 276-294 (1979)

[7] U. Baumann, On Edge-Hamiltonian Cayley Graphs, Discrete Applied Math-ematics 51: 21-37 (1994)

[8] K. A. Berman, Proof of a Conjecture of Haggkvist on Cycles and IndependentEdges, Discrete Mathematics 46: 9-13 (1983)

[9] Y. C. Chen, Z. Furedi, Note Hamiltonian Kneser Graphs, Combinatorica 22:147-149 (2002)

[10] Y. C. Chen, Kneser Graphs are Hamiltonian for n ≥ 3k, Journal of Combi-natorial Theory B 80: 69-79 (2000)

[11] Y. C. Chen, Triangle-free Hamiltonian Kneser Graphs, Journal of Combina-torial Theory B 89: 1-16 (2003)

[12] B. L. Chen, K. W. Lih, Hamiltonian Uniform Subset Graphs, Journal of Com-binatorial Theory B 42: 257-263 (1987)

[13] V. Chvatal, Tough Graphs and Hamiltonian Circuits, Discrete Mathematics306: 910-917 (2006)

59

Page 62: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

BIBLIOGRAPHY

[14] V. Chvatal, P. Erdos, A Note on Hamiltonian circuits, Discrete Mathematics2: 111-113 (1972)

[15] R. C. Compton, S. G. Williamson, Doubly adjacent Gray Codes for the sym-metric group, Linear and Multilinear algebra 35: 237-293 (1993)

[16] S. J. Curran, J. A. Gallian, Hamilton Cycles and Paths in Cayley Graphs andDigraphs - a Survey, Discrete Mathematics 156: 1-18 (1996)

[17] C. Godsil, G. Royle, Algebraic Graph Theory, Springer (2001)

[18] K. Heinrich, W. D. Wallis, Hamiltonian cycles in certain graphs, Journal ofthe Australian Mathematical Society A 26: 89-98 (1978)

[19] J. v. d. Heuvel, Hamilton Cycles and Eigenvalues of Graphs, Linear Algebraand its Applications 226-228: 723-730 (1995)

[20] J. v. d. Heuvel, B. Jackson, On the Edge Connectivity, Hamiltonicity, andToughness of Vertex Ttransitive Graphs Journal of Combinatorial Theory B77: 138-149 (1999)

[21] W. Holsztynski, R.F.E. Strube, Paths and Circuits in Finite Groups, DiscreteMathematics 22: 263-272 (1978)

[22] B. Jackson, Hamilton Cycles in Regular 2-Connected Graphs, Journal of Com-binatorial Theory B 29: 27-46 (1980)

[23] M. Jiang, F. Ruskey, Determining the Hamilton-connectedness of certain Ver-tex Transitive Graphs, Discrete Mathematics 133: 159-169 (1994)

[24] D. Jungreis, E. Friedman, D. Witte, Caylay Graphs on Groups of Low Orderare Hamiltonian, Preprint

[25] K. Keating, The Conjuction of Caylay Digraphs, Discrete Mathematics 42:209-219 (1982)

[26] J. B. Klerlein, Hamiltonian Cycles in Cayley Color Graphs, Journal of GraphTheory 2: 65-68 (1978)

[27] V. L. Kompel’makher, V. A Liskovets, Sequential Generation of Arrangementsby Means of a Basis of Transpositions, Kibernetica 3: 17-21 (1975)

[28] K. Kutnar, D. Marusic, Hamiltonicity of Vertex Transitive Graphs of Order4p, Preprint (2006)

[29] J. H. v. Lint, R. M. Wilson, A Course in Combinatorics, Cambridge Univer-sity Press: 536-541 (1992)

[30] L. Lovasz, Combinatorial Problems and Exercises, Amsterdam (1970)

60

Page 63: A study of necessary and sufficient conditions for vertex ... · highly symmetric graphs, or vertex transitive graphs. This thesis can be divided into two parts. In the first part

BIBLIOGRAPHY

[31] M. Mather, The Rugby Footballers of Croam, Journal of Combinatorial TheoryB 20: 62-63 (1976)

[32] G. H. J. Meredith, E. K. Lloyd, The Hamiltonian Graphs O4 to O7, Combi-natorics (1972)

[33] G. H. J. Meredith, E. K. Lloyd, The Footballers of Croam, Journal of Combi-natorial Theory B 15: 161-166 (1973)

[34] B. Mohar, A Domain Monotonicity Theorem for Graphs and Hamiltonicity,Discrete Applied Mathematics 36: 169-177 (1992)

[35] B. Mohar, The Laplacian Spectrum of Graphs, Graph Theory, Combinatoricsand Applications 2: 871-898 (1991)

[36] I. Pak, R. Radoicic, Hamiltonian Paths in Cayley Graphs, Preprint (2004)

[37] W. T. Tutte, A Non-Hamiltonian Graph, Canadian Mathematical Bulletin 3:1-5 (1960)

[38] W. T. Tutte, A theorem on planar graphs, Transactions of the AmericanMathematical Society 82: 99-116 (1956)

[39] D. B. West, Introduction to Graph Theory, Prentice Hall, second Edition(2001)

[40] A. T. White, Ringing the cosets, American Mathematics Monthly 94-8: 721-746 (1987)

[41] D. S. Witte, On Hamiltonian Circuits in Cayley Diagrams, Discrete Mathe-matics 38: 99-108 (1982)

[42] D. S. Witte, J.A. Gallian, A survey: Hamiltonian Cycles in Cayley Graphs,Disrete Mathematics 51: 293-304 (1984)

[43] D. Witte, Cayley Digraphs of Prime-power Order are Hamiltonian, Journalof Combinatorial Theory B 40: 107-112 (1986)

61