On Saturated Graphs

Saturation NumbersEdge Spectrum of CliquesWhat is known for Trees?

On Saturated Graphs

Ron Gould

April 10, 2010

Ron Gould On Saturated Graphs


Definition

A graph G is said to be H-saturated provided it does not containH as a subgraph, but the addition of any new edge creates at leastone copy of H in G .



A classic question in graph theory is the following:

Question

What is the maximum number of edges in an H saturated graph

on n vertices?

This maximum is denoted ex(n, H) and called the extremal

number for H.



The fundamental result by Turan, in 1940.

Theorem

There is a unique extremal graph on n vertices that is Kp+1-free,

namely Tn,p.

The graph Tn,p is a balanced p-partite graph on n vertices.Of course this means ex(n, K3) = ⌊n2/4⌋.



Erdos - Stone - Simmonovits Theorem

Theorem

limn→∞

ex(n; G )

n2=

1

2(1 −

1

χ(G ) − 1).

Also, Stability Theorems!



Extremal graphs are saturated and very important. But, there canbe other H-free graphs that are also H-saturated.

For a graph H, we will denote the minimum size of anH-saturated graph by sat(n, H).

Problem

Determine sat(n, H) for any graph H.



In 1964 Erdos, Hajnal and Moon determined that:

sat(n, Kt) = (t − 2)(n − 1) −(

t−22

)

.

This arises from the graph Kt−2 + Kn−t+2, where + denotes join.



In 1964 Erdos, Hajnal and Moon determined that:

sat(n, Kt) = (t − 2)(n − 1) −(

t−22

)

.

This arises from the graph Kt−2 + Kn−t+2, where + denotes join.

Kt−2 (n − t + 2)K



In particular, this means that for a triangle, the graph is just a star(with n − 1 edges), thus sat(n, K3) = n − 1.



In particular, this means that for a triangle, the graph is just a star(with n − 1 edges), thus sat(n, K3) = n − 1.

star



Thus we have seen that the extremal number for triangles is O(n2)while the saturation number is O(n).

This is no accident (as it is the case for most graphs).

Kaszonyi and Tuza provided a very general upper bound onsaturation numbers and used it to show the following:



Kaszonyi and Tuza, 1986

Theorem

For every graph F there exists a constant c such that

sat(n, F ) < cn.



Kaszonyi and Tuza also considered matchings

Theorem

sat(tK2, n) = 3t − 3 for n ≥ 3t − 3.



multiple copies of Kt

with Faudree, Ferrara, and Jacobson we were able to show acommon generalization of EHM for cliques and KT for matchings:

Theorem

sat(tKp, n) = |E (G (n, p, t))|

= (t − 1)

(

p + 1

2

)

+

(

p − 2

2

)

+ (p − 2)(n − p + 2).



Now consider the graph G (n, p, t):

(t − 1)Kp+1

Kp−2

Kn−pt−t+3

. . .



Problems!

Let F be a family of graphs.Then ex(n;F) satisfies:



Problems!


ex(n;F) ≤ ex(n + 1;F)



Problems!


ex(n;F) ≤ ex(n + 1;F)

If F1 ⊂ F then ex(n;F1) ≥ ex(n;F)



Problems!


ex(n;F) ≤ ex(n + 1;F)


If H is a subgraph of G , then ex(n; H) ≤ ex(n; G )



Problems!


ex(n;F) ≤ ex(n + 1;F)


If H is a subgraph of G , then ex(n; H) ≤ ex(n; G )

These do not hold for saturation numbers in general!



Example of 1

Note that sat(P4, n = 2k) = k while sat(P4, n = 2k − 1) = k + 1.

. . . . . .

2k 2k−1



Example for 3. Consider K4 and the supergraph H obtained byattaching an edge to K4. We know that sat(n, K4) = 2n − 3.But for H we have:

here n = 4m and size = 6m

hence size = 3n/2



In fact, Barefoot, Casey, Fisher, Fraughnaugh and Harary,(1995) showed that:

Theorem

For n ≥ 5, there exists a K3-saturated graph of order n with m

edges if and only if it is complete bipartite or

2n − 5 ≤ m ≤ ⌊ (n−1)2

4 ⌋ + 1.



with K. Amin and J. Faudree

Recently, we were able to generalize this result for all t ≥ 3.

Theorem

For n ≥ 3t + 5, there is a Kt-saturated graph G of order n with m

edges if, and only if, G is complete (t − 1)-partite or

(t − 1)(n − t/2) − 2 ≤ m ≤ ⌊ (t−2)n2−2n+(t−2)

2(t−1) ⌋ + 1.



K4-saturated graphs

Theorem

If G is a K4-saturated graph of order n and

if κ(G ) = 2, then G ∼ K1,1,n−2,

if κ(G ) = 3, then G ∼ K1,2,n−3 or G ∼ W (1, t, 1, r , s).



K4-saturated graphs

Theorem

If G is a K4-saturated graph of order n and

if κ(G ) = 2, then G ∼ K1,1,n−2,

if κ(G ) = 3, then G ∼ K1,2,n−3 or G ∼ W (1, t, 1, r , s).

t

rs



Examples

n−1

2n−3 n2

2n−5

3n−8

+1−2 n2

n

4

n2

−n + 1

3+1

isolated values

2n

4

K 3

K4

no values fromcomplete bipartite graphs

3



Proof (rough sketch)The extreme values are clear from Turan and EHM.To show the upper end of the interval is an easy application of aresult of Hanson and Toft.To show the lower end of the interval was harder. It requiredsequence of lemmas using the chromatic number of the graphalong with the fact it was Kt-saturated and had a certain minimumdegree. Then you back the graph into either having enough edgesor in some cases, the wrong chromatic number and therefore itcould be eliminated.Doing this for χ being t − 1, t and t + 1 allowed us to then use aresult of Duffus and Hanson to handle the remaining cases.



Sir William Rowan Hamilton - (1805-1865)



What about trees?Note paths and stars were considered by Kaszonyi and Tuza,

1986.

Theorem

sat(n, Sk+1) =

{

(

k2

)

+(

n−k2

)

if k + 1 ≤ n ≤ k + k2

⌈k−12 n − k2

8 ⌉ if k + k2 ≤ n.



Theorem

1. For n ≥ 3, sat(n, P3) = ⌊n/2⌋.2. For n ≥ 4,

sat(n, P4) =

{

n/2 n even

(n + 3)/2 n odd.

3. For n ≥ 5, sat(n, P5) = ⌈5n−46 ⌉.

4. Let

ak =

{

3 · 2t−1 − 2 if k = 2t

4 · 2t−1 − 2 if k = 2t + 1.

If n ≥ ak and k ≥ 6 then

sat(n, Pk) = n − ⌊n

ak

⌋.



Kaszonyi and Tuza showed that, of all trees on k vertices, theone with largest saturation number is the star and it is unique.We (FFGJ) show that the unique tree on k vertices with smallest

saturation number is T ∗

k , a star with a single subdivided edge.

Theorem

For any tree Tk of order k ≥ 5 and any n ≥ k + 2,

sat(n, Tk) ≥ n − ⌊n + k − 2

k⌋.

Moreover, T ∗

k is the only tree attaining this minimum for all n.



Let δ2(T ) be the second smallest degree in T .

Theorem

If T is a tree of order at least 5 such that δ2(T ) = d, then

sat(n, T ) ≥ d−12 n − d2

8 .

Corollary

Every tree T is a subtree of a tree T ′ such that sat(n, T ′) ≥ αn

for any constant α and n sufficiently large.

Theorem

Every tree T is the subtree of a tree T ′ such that sat(n, T ′) < n.



Cl -saturated graphs of minimum size

l sat(n, Cl) n ≥ Reference

3 = n − 1 3 EHM

4 ⌊3n−52 ⌋ 5 OT

5 ⌈10n−107 ⌉ 21 Chen

6 ≤ 3n2 11 Many

7 ≤ 7n+125 10 Many

8,9,11,13,15 ≤ 3n2 + l2

2 2l Thm 3

≥ 10 and ≡ 0 mod 2 ≤(

1 + 2l−2

)

n + 5l2

4 3l Thm 1

≥ 17 and ≡ 1 mod 2 ≤(

1 + 2l−3

)

n + 5l2

4 7l Thm 2

n ⌊3n+12 ⌋ 20 Many

Table: A Summary of Results for sat(n,Cl )


On Saturated Graphs

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