Graph minor theory

Basic concepts and DefinitionsAchievements and remarkable works

Extremal problems and resultsOther approaches

Signed graphs and minorsSketch of proofs

Graph Minor Theory

Bahman Ghandchi

Institute for Advanced Studies in Basic Sciences (IASBS)

ICTP, TriesteSeptember 11, 2012

Institute for Advanced Studies

in asic Sciences

Gava Zang, anjan, Iran

Bahman Ghandchi Graph Minor Theory




Table of contents

1 Basic concepts and DefinitionsEdge contractionMinorTopological minor

2 Achievements and remarkable worksInteresting works and resultsHadwiger conjectureHajos conjectureWagner conjecture

3 Extremal problems and resultsMader works and related resultsKs,t Minor free graphs

Extremal results containing Girth

4 Other approachesProbabilistic methods and Hajos conjecture

5 Signed graphs and minorsSigned graphs

6 Sketch of proofsKtK2,tKs,t





Edge contractionMinorTopological minor

Edge contraction

Definition (edge contraction)

Let e = xy be an edge of a graph G = (V, E). By G/e we denotethe graph obtained from G by contracting the edge e into a newvertex ve , which becomes adjacent to all the former neighbours ofx and of y.Formally G/e is a graph (V ,E ) with vertex setV := (V \{x , y}) {ve}(where ve is the new vertex, i.e. ve / V E ) and edge set:E := {vw E | {x , y} {v ,w} = } {vew | xw E\ {e} or yw E\ {e} } .






Minor

Definition (minor)

A graph H is a minor of a graph G if a graph isomorphic to H canbe obtained from a subgraph of G by contracting edges[2].

Definition (another minor definition)

Any graph H that can be produced from G by successiveapplication of these reductions is called a minor of G:(a) delete an edge,(b) contract an edge,(c) delete an isolated node.






Topological minor

Definition (topological minor)

A graph H is called a topological minor of a graph G if asubdivision of H is isomorphic to a subgraph of G.

Theorem

every topological minor of a graph is also its (ordinary) minor.

Theorem

every minor with maximum degree at most 3 of a graph is also itstopological minor.





Interesting works and resultsHadwiger conjectureHajos conjectureWagner conjecture

Interesting works and results

During the past decades graphs minors theory has been developedso well that one can name the series of papers by Robertson andSeymour(Graph Minors I...XXIII) as the most important work ingraph theory. many interesting results and approaches are present,just to name a few we have the following short list:

algorithmic aspects: finding O(n3) algorithm for solvingk-Disjoint path problem for fixed k.

algorithmic aspects: if we bound tree-width of instances ofmany NP-hard problems we can solve those in poly-time.

Wagner conjecture:For every minor-closed family of graphsthe set of forbidden minors is finite.

probabilistic methods: Hajos conjecture disproof and recentworks in proving it for large girth.






Hadwiger conjecture

One of the most important and challenging open problems ingraph theory is Hadwigers conjecture:

Conjecture (Hadwiger 1943)

for every integer r > 0 and every graph G:(G ) r G Kr

key facts:

this conjecture is true for r < 7 and still open for greatervalues.

as (Kt,t) = 2 Kt,t Kt nothing can be said conversely.






Hajos conjecture

The Hajos conjecture is a strengthened version of Hadwigerconjecture which states:

Conjecture (Hajos)

for every integer r > 0 and every graph G:(G ) r GtKr

Key facts about this conjecture:

Hajos conjecture has been failed in general.

conjecture is true for r 4 and false for r 7 and cases 5 &6 are still open.

Erdos has showed with probabilistic methods that almostevery graph which is large enough is a counter example forthis conjecture.






Wagner conjecture

Wagner conjecture or as it is known today (Robertson Seymourtheorem) is one of the most important works in graph theory inpast decades. the theorem states:

Theorem

For every minor-closed family of graphs the set of forbidden minorsis finite.

this theorem generalizes the planar graphs theorem in which wehave K5 &K3,3 as forbidden minors.a variation of this theorem is for being linklessly embeddable:

Theorem

A graph is linklessly embeddable if and only if it does not containany of the seven graphs in Petersen family as a minor.






Wagner conjecture

The Petersen family






Wagner conjecture

the following theorem is equivalent to this theorem which statesthe set of all finite graphs with minor relation is well quasi ordered

Theorem

For every infinite sequence G1,G2, ... of graphs, there exist distinctintegers i < j such that Gi is a minor of Gj .

the proof of Wagners conjecture is one the main result from seriesof 23 papers named Graph Minors I to Graph Minors XXIIIpublished by Robertson and Seymour from 80s to 2004. theseworks are considered as one the most important projects in graphtheory.





Mader works and related resultsKs,t Minor free graphsExtremal results containing Girth

Mader works

Mader has proved that for every graph H there is a constant CHsuch that every graph G not containing H as a minor satisfies|E (G )| CH |V (G )|, but determining the best possible constantCH for a given graph H is a question that has been answered forvery few graphs H.In fact Mader has shown that:

Theorem (Mader 1967)

There is a function h : N N such that every graph with averagedegree at least h(r) contains Kr as a topological minor for everyr N.

The function obtained in this theorem is h(r) = 2r(r1)/2.as we will see in the following sections this bound is so loose.






Bounds for Kr (topological)minor free graphs

Two bounds for Kr (topological)minor free graphs have been foundand these bounds are sharp up to a constant c as a function of r.

Theorem (Bollobas & Thomason and independently Komlos &Szemeredi)

There exists a c R such that, for every r N, every graph G ofaverage degree d(G ) > cr 2 contains Kr as a topological minor.

Theorem (Kostochka 1982; Thomason 1984[5])

There exists a c R such that, for every r N, every graph G ofaverage degree d(G ) > cr

log r contains Kr as a minor.






K1,t minor free graphs

It is easy to see that every n-vertex graph with more than12 (t 1)n edges contains K1,t as a minor (indeed, as a subgraph),and if t divides n then there is an n-vertex graph with exactly12 (t 1)n edges with no K1,t minor (the disjoint union of n/tcopies of Kt ).The extremal example for above statement is not connected andthe answer when we restrict ourselves to connected graphs isdifferent.

Theorem (G. Ding, P. Seymour and T. Johnson, 2001)

Let t 3 and n t + 2 be integers. If G is an n-vertex connectedgraph with no K1,t minor, then |E (G )| n + 12 t(t 3)and for all n, t this is best possible.






K2,t minor free graphs

Theorem (Chudnovsky, Reed and Seymour 2011[1])

Let t 2, and let G be a graph with n > 0 vertices and with noK2,t minor. Then

|E (G )| 12

(t + 1)(n 1)

.

key facts:

Myers had previously proved the theorem for all t 1029.the bound is best possible when n - 1 is a multiple of t.

for n = 32 t the bound is about12 tn but the best known result

is 512 tn.






Ks,t minor free graphs

More generally, what is the maximum number of edges in graphswith no Ks,t minor when s > 1? If we take a graph eachcomponent of which is a clique of size t, and add s 1 morevertices each adjacent to all others, then the resulting n-vertexgraph has no Ks,t minor, and the number of edges is :(t + 2s + 3)(n s + 1)/2 + (s 1)(s 2)/2.Key facts:

Kostochka and Prince have a proof of this for all sufficientlylarge t.

it is open for s = 4, 5.

for s 6 Kostochka and Prince have counterexamples.Kostochka and Prince proved the following:






Ks,t minor free graphs

Theorem (A. Kostochka and N. Prince[3])

Let s,t be positive integers with t > (240s log2 s)8s log2 s+1. Then

every graph with average degree at least t + 3s has a Ks,t minor,and there are graphs with average degree at least t + 3s 5

s

that do not have a Ks,t minor.

Theorem (A. Kostochka[4])

let s and t be positive integers such that

t > t0(s) := max 415s2+2, (240s log2(s))

8s log2 s+1 (1)

then every (s+t)-chromatic graph has a ks,t-minor






Minor and Girth

There are many results in minors theory which uses girth of agraph just to name a few we have these two theorems. theinteresting fact is the second theorem shows that only big enoughgirth can force certain minors with high minimum degree.

Theorem (Mader 1997)

For every graph H of maximum degree d 3 there exists aninteger k such that every graph G of minimum degree at least dand girth at least k contains H as a topological minor.

Theorem (Thomassen 1983)

Given an integer k, every graph G with girth g(G ) 4k 3 and(G ) 3 has a minor H with (H) k.





Probabilistic methods and Hajos conjecture


Using the chernoff bound we have the following result:

Theorem (Erdos, Fajtlowicz)

For n sufficiently large there exists graphs with chromatic numberat least n2log2n

and no topological minor of K8n.

majority of large graphs are counterexamples.

we should have n 230

Hajos conjecture is true for large girth.






Theorem (Kuhn, Osthus 2006)

Let r 1 be a natural number. Every graph of minimum degree atleast r and girth at least 27 contains a subdivision of Kr+1 .

Its a fact that every kchromatic graph has a subgraph withminimum degree at least k 1. So combining this fact and theabove theorem we have a weakened version of Hajos conjecturewhich is true.

Theorem

If G is graph with girth(G ) 27 and (G ) = k the G has atopological minor of Kk .





Signed graphs

Signed graphs

A graph G with a subset of its edges S E (G ) is called a sigendgraph. Edges that are in S are negative and edges which are notare positive. A resigning operation at any vertex is allowed so asigned graph is not a single graph but a family of graphs.

(G ,S) (2)





Signed graphs

Signed minors

A signed graph (G1, S1) is called a signed minor of (G2,S2) if itcan be obtained from (G2, S2) from a sequence of followingoperations:

Vertex deletion

Edge deletion

Resigning

Contraction on positive edges





KtK2,tKs,t

Kt minor free graphs sketch of proof

The main result discussed here is by Thomason and is as follows:

Theorem (Kostochka 1982; Thomason 1984[5])

There exists a c R such that, for every r N, every graph G ofaverage degree d(G ) > cr

log r contains Kr as a minor.





KtK2,tKs,t


Suppose that the function below is defined for every integer pwhere G is a graph:

c(p) = inf{c |G ; |E (G )| c|V (G )| G Kp}, (3)

It will be shown that for large p we have:

c(p) 2.68p

log p (4)

Define real number by = 1 + log 2; then = 2.678





KtK2,tKs,t


Lemma

let r = {G ||V (G )| r 2(G ) brc 3}, where r is real andr 3 is an integer. if G r implies G Kp2 then c(p) r

First we choose a graph H1 which is minor minimal in the set

{G ||E (G )| r |V (G )|}. (5)

We will show that G Kp.Then a vertex of minimum degree such as z is chosen in H1. LetH2 = H1|N(z) it is easy to show that (H2) r .next step is to show H2 Kp1.





KtK2,tKs,t


Let set D and function f on set of finite graphs be defines asfollows:

f (G ) = r |V (G )|(log(|V (G )|/r) + 1)/2. (6)

andD = {G ||V (G )| r |E (G )| f (G )}. (7)

By definition of H2 D. Let H be a minor of H2 which is minorminimal in the set D.Next we choose a vertex u V (H) such that dH(u) = (H) andput G = H|N(u).G satisfies both |V (G )| r and 2(G ) brc 3 so G D.





KtK2,tKs,t


therefore:

G Kp2 H2 H Kp1 H1 Kp (8)

As desired.the proof continues with some technical lemmas to help us provethat indeed G D implies G Kp2. The main lemma which isneeded here is as follows:

Lemma

Let G be a graph of order n with (G ) n+m2 for some

non-negative integer m n 3. Then there are at least(

m + 3

k

)k subsets of G each dominating all but some b2knc vertices ofG .





KtK2,tKs,t


The main theorem is stated as follows:

Theorem

Let r = r(p) be a function of p such that r is an integer and suchthat there is an integer k satisfying both r (p 3)l k 5 and

(p 3)(b2krc

k

) 0 vertices and with noK2,t minor. Then

|E (G )| 12

(t + 1)(n 1)

.





KtK2,tKs,t

K2,t minor free graphs sketch of proof

Fix t 2 and suppose the theorem is false for that value of t. Sothere is a minimal counterexample, that is, a graph G with thefollowing properties:

G has no K2,t minor.

|E (G )| > 12 (t + 1)(|V (G )| 1).|E (G )| > 12 (t + 1)(|V (G

)| 1) for every graph G with noK2,t minor and |V (G )| < |V (G )|.

Since |E (G )| > 12 (t + 1)(n 1) it follows that n t + 2.





KtK2,tKs,t


The proof is technical and so long but the approach is outlinedhere:

Proving some basic lemmas about G containing connectivity.

Discovering some facts about size of neighbour set of littlesubsets.

Small t cases are handled.

An edge is found with large neighbourhood.





KtK2,tKs,t


The proof is long and technical but some major lemmas are listedbelow:

Lemma

For every edge xy of G there are at least 12 t xyjoins andconsequently every vertex has degree at least 12 t.

Lemma

For every two non adjacent vertices x , x there are at least 3xx joins, and so G is 3-connected.

Lemma

For every vertex v of G there are at least two vertices non adjacentto v.





KtK2,tKs,t


Lemma

G is 5-connected and so t 6.

Lemma

Let W V (G ) be connected with |W | 2. If t 11 then|N(W )| t + 3.

Lemma

Let W V (G ) be connected with |W | 2. Then|N(W )| t + 3, and if t 13 then |N(w)| t + 2





KtK2,tKs,t


then in a series of deductions an edge is found where the size ofits neighbour set contradicts the bound in last lemma:Let w be a vertex of maximum degree t + s. Let A = N(w) andB = M(w). Let e1 be the number of edges between A and B ande2 the number of edged with both end in B. Let (B) be thenumber of components of B and A0 be the set of vertices in Awith no neighbour in B. Let X be the set of all vertices in A withdegree t + 1. Let d = 2 if t 13 and d = 3 otherwise.Now we will complete the proof with deducing a contradiction in14 steps.





KtK2,tKs,t

K2,t minor free graphs sketch of proof I

1 Every vertex in A has at most 4 s neighbours in B, and atmost 3 s if t 13.

2 Every vertex in B has at least max(3, 12 t + s 2) neighboursin A, and at least max(3, 12 t + s 1) if t 13.

3 Every vertex in A has at most t s neighbours in A.4 s 2.5 If s = 2, then t 14 and e2 1 and |B| 3.6 If s = 2 then |B| = 2.7 s = 1, and therefore every vertex in G has degree at most

t + 1, and t |B| 1.





KtK2,tKs,t

K2,t minor free graphs sketch of proof II

8 |A0|+ (B) 3, and for every component of B, at most t 2vertices in A have neighbours in C .

9 |X |+ e1 + 2e2 (t + 1)|B|+ 1, and |X |+ |A0| t + 1 and so2e2 (t + 1)(|B| d 1) + (d + 1)|A0|+ 1

10 |B| 5, and if t 13 then |B| 4.11 |B| 4.12 |B| 3.13 there is a dominating edge.

14 At most two vertices in A have more than one neighbour in B.





KtK2,tKs,t

K2,t minor free graphs sketch of proof III

Then it is shown that B is isomorphic to K3.By (8) and (14) we have:

e1 t + 2 and |A0| 3. (12)

By (9) we have

(t + 1 |A0|) + e1 + 2e2 (t + 1)|B|+ 1. (13)

And so

(t 2) + (t + 2) + 6 3(t + 1) + 1. (14)

Which is t 2, and it is a contradiction. So we have our desiredresult:





KtK2,tKs,t

K2,t minor free graphs sketch of proof IV

Theorem (Chudnovsky, Reed and Seymour 2011[1])

Let t 2, and let G be a graph with n > 0 vertices and with noK2,t minor. Then

|E (G )| 12

(t + 1)(n 1)

.





KtK2,tKs,t

Ks,t minor free graphs sketch of proof

The main theorem in this section is stated below. Let K s,t be thegraph obtained form Ks,t by adding all edges between the verticesof partite set of size S .



t > t0(s) := max{415s2+2, (240s log2(s))

8s log2 s+1} (15)






KtK2,tKs,t


Kostochka and Prince previously proved the following theorem:

Theorem

Let t 6500. Then every (3 + t)chromatic graph has aK 3,tminor and every (2 + t)chromatic graph has a K 2,tminor.

Proof is by induction on s. suppose that the theorem is proved forall s < s and t > t0(s

). It is enough to consider s 4.Let G0 be a counter example for s and some t > t0(s) which isminimal with respect to |V (G )|+ |E (G )|.Here is a list of important lemmas and theorems which are used inproving this theorem:





KtK2,tKs,t


Theorem (1)

Let s,t be positive integers with t > (240s log2 s)8s log2 s+1. Then

every graph with average degree at least t + 3s has a Ks,t minor,and there are graphs with average degree at least t + 3s 5

s

that do not have a Ks,t minor.

Lemma (2)

Let s, t be positive integers such that t > t0(s). Let G be a15s2connected graph. Suppose that G contains a vertex subsetU with:

t + 700s3 ln t |U| 3t, (16)

such that (G |U) t/(4s + 1). Then G has a K s,tminor.





KtK2,tKs,t


Lemma (3)

n0 2(s + t) 1

Lemma (4 Seymour)

Let k be a non negative integer. If v V (G0) andd(v) = s + t 1 + k, then (G0[N(v)]) k + 1

Lemma (5)

Let k be a non negative integer. If v V (G0) andd(v) = s + t 1 + k, then there exists a subset Y (v) N(v) suchthat (G0[Y (v)

{v}]) tk+1





KtK2,tKs,t


Lemma (6)

If t 4x+s , then the connectivity of G0 is greater than x.

By theorem 5 we have

vV (G0)d(v) < (t + 3s)(n0 s + 1). (17)

Let L be the set of vertices of G0 with degree less than t + 5s 1.so we have:

|L|(t + s 1) + (n0 |L|)(t + 5s) < (t + 3s)(n0 s + 1). (18)

which is

|L| 2s4s + 1

n0 +s 1

4s + 1t. (19)





KtK2,tKs,t


Since t > 415s2+s and by lemma 6, G0 is 15s

2connected.By lemma 3 we have n0 2t + 2s 1.If n0 3t, then G0 with U = V (G0) satisfies in lemma 1 andcontains a K s,tminor which is contradiction so:

n0 3t + 1. (20)

Thus for s 3 we have:

|L| 613

3t +2

13t =

20

13t. (21)

Using lemma 5 we find a subset U that together with G0 satisfiesin lemma 1 so G0 has a K

s,tminor which is contradiction.





KtK2,tKs,t


So as desired we have the result:



t > t0(s) := max{415s2+2, (240s log2(s))

8s log2 s+1} (22)






references I

M. Chudnovsky, B. Reed, and P. Seymour.The edge-density for k2,t minors.Journal of Combinatorial Theory, Series B, 101(1):18 46,2011.

K. Kawarabayashi and B. Mohar.Some recent progress and applications in graph minor theory.Graphs Combin, 23, 2007.

A. Kostochka and N. Prince.On ks,t minors in graphs of given average degree.Discrete Math., (308):44354445, 2008.





references II

A. Kostochka and N. Prince.On ks,t minors in (s+t)-chromatic graphs.Journal of Graph Theory, (65):343350, 2010.

A. Thomason.The extremal function for complete minors.J. Combin. Theory Ser. B, (81):318338, 2001.

Basic concepts and DefinitionsEdge contractionMinorTopological minorAchievements and remarkable worksInteresting works and resultsHadwiger conjectureHajos conjectureWagner conjectureExtremal problems and resultsMader works and related resultsKs,t Minor free graphsExtremal results containing GirthOther approachesProbabilistic methods and Hajos conjectureSigned graphs and minorsSigned graphsSketch of proofs Kt K2,t Ks,t

Graph minor theory

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