The Turán number of sparse spanning graphs

Raphael Yuster

joint work with

Noga Alon

Banff 2012

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ex(n,H) is the maximum number of edges in a graph of order n not containing a subgraph isomorphic to H.

Ore’s result states that:

Recently, Ore's theorem has been generalized to the setting of Hamilton cycles in k-graphs:

• Let Cn(k,t) denote the (k,t)-tight cycle of order n.

• [Glebov, Person & Weps – 2012] determined ex(n,Cn(k,t)) (for n suff. large).It is of the form where P is a specific fixed (k-1)-graph.

[Ore – 1961]A non-Hamiltonian graph of order n has at most edges.

t

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It is natural to try to extend Ore's result to spanning structures other than just Hamilton cycles (in both the graph and hypergraph settings).

Suppose that H is a k-graph of order n and with, say, bounded max degree.

It is natural to suspect that for n sufficiently large,

where L is a set of (k-1)-graphs that depending on neighborhoods in H.

A conjecture raised in [GPW – 2012] asserts that it suffices to take L to be the set of links of H. (example: the links of a Cn(3,1) are L={K2 , 2K2} ).

Observe: the conjecture holds for both Ore's result and its aforementioned generalization to Hamilton cycles in hypergraphs (in fact, with equality).

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In the graph-theoretic case, the link of a vertex is just a set of singletons whose cardinality is the degree of the vertex.

In this case, the aforementioned conjecture states that:

if H is a graph of order n with mindeg δ>0 and bounded maxdeg, then

assuming n is sufficiently large (note: we trivially cannot do better).

Main result: This is true in a strong sense (no need for bounded maxdeg):Theorem 1:For all n sufficiently large, if H is any graph of order n with no isolated vertices and , then

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• Proof actually works for all n > 10000.

• The constant 40 cannot be improved to less than .hence the bound on the maximum degree is optimal:

• Take H with n=k(k+6)/2+1 vertices, consisting of k disjoint cliques of size (n-1)/k each, and another vertex connected to δ ≤ (n-1)/k-1 vertices of the cliques.

• Clearly, Δ(H)=(n-1)/k and δ(H)=δ .

• H has no independent set of size k+2.

• Hence, if G is Kn - Kk+2, then H is not a spanning subgraph of G.

• However, G has more than edges.

For all n sufficiently large, if H is any graph of order n with no isolated vertices and , then

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A counter-example to the conjecture of [GPW – 2012], already for 3-graphs:

Proposition 2:Let s be a large integer, n=1+5s and let V1… Vs {x}

where |Vi|=5.

Let H be the 3-graph on V where each Vi forms a K5(3) and x is contained in a unique edge {x,u,v} with u,v in V1.

Then ex(n-1,L(H)) = 0 but:

Proof: Take T to be U1 U2 U3 {x,y} where |Ui|=(n-2)/3.The edges are all the triples of U1U2U3 and all triples {x,ui,uj} ,{y,ui,uj}.T does not contain H because the links of x and y are 3-colorable so do not lie in a K5(3) . The result follows since T has edges.

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Proof preliminaries

We say that G and H of the same order pack, if H is a spanning subgraph of the complement of G.

Let H=(W,F) be a graph with n vertices and with .

Let G=(V,E) be any graph with n vertices and n-δ-1 edges, where δ=δ(H).

It suffices to prove that G and H pack.

Equivalently, a bijection f : V W such that (u,v) E (f(u), f(v)) F.

Let V={v1,…,vn} where d(vi) ≥ d(vi+1) .

Observe: d(v1) ≤ n-δ-1 d(v2) ≤ n/2 d(vi) ≤ 2n/i

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We need the following independent sets of G, one for each vi :

S1 consists of non-neighbors of v1 that have small degree (less than 2n1/2)

Si consists of non-neighbors of vi that have very small degree (at most 50)

Each Si is chosen with maximum cardinality, under this restriction.

It is not difficult to show that |S1| ≥ δ and |Si| ≥ n/7.

vi

N(vi)

Si

Random subsets of Si have whp some useful properties for our embedding:

Let Bi be a random subset of Si where each vertex is chosen with prob. n-1/2

Lemma 1: Whp, all the Ci are relatively small (less than 4n1/2)

the first few Di are relatively large (at least 0.05n1/2 for i=2,…,n1/2)

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Proof outline

The construction of the bijection f : V W is done in four stages.

At each point of the construction, some vertices of V are matched to some vertices of W while the other vertices of V and W are yet unmatched.Initially, all vertices are unmatched.

We always maintain the packing property:

for two matched vertices u,v V with (u,v) E we have (f(u), f(v)) F.

Thus, once all vertices are matched, f defines a packing of G and H.

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Stage 1.

We match v1 (a vertex with maximum degree in G) with a vertex w W having minimum degree δ in H.

As N(w)= δ and since |S1| ≥ δ , we may match an arbitrary subset B1 of δ vertices of S1 with N(w).

Observe that the packing property is maintained since B1 is an independent set of non-neighbors of v1 .

Note that after stage 1, precisely δ+1 pairs are matched.v1

N(v1)

|S1| ≥ δ

w

|N(w)|=δ

S1

N(w)

G

HOther vertices

Stage I1.

This stage consists of iterations i=2,…,k where at iteration i we match vi and some subsets of Bi with a corresponding set of vertices of H.We do this as long as d(vi) ≥ 2n1/2 (hence k ≤ n1/2).

We make sure that after each iteration i, the following invariants are kept:

1. After matching vi with some vertex w=f(vi) of H, we make sure that all neighbors of w in H are matched to vertices of Bi .

2. Any matched vertex of G other than {v1,…,vi} is contained in some Bj where j ≤ i.

3. The number of matched vertices after iteration i is at most i((H)+1).

These invariants clearly hold after stage 1. 11

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vi

Observe: it is really yet

unmatched

matched vj j < iOther matched

neighborsunmatched neighbors

X Y Z Si Non-neighbors

Bi

|X|<i<k<n1/2 Y j<iBj

Y N(vi), Y Ci

|Y|<5n1/2Lemma

1

T

The matches of X Y in H

|T|=|X|+|Y| <6n1/2

Q

non-neighbors of T

|Q| ≥ n-|T|Is there an unmatched vertex in Q?

Yes! only (i-1)(|+1) matched so far

w

Unmatchedneighbors of w

R|R| ≤ ≤ n1/2/40

Di =Bi-j<iBj

|Di| ≥ n1/2/20, lemma 1

G

H

Stage I1I.

We are guaranteed that the unmatched vertices of G have degree ≤ 2n1/2.

By the third invariant of Stage 2, the number of unmatched vertices of G is still linear in n (at least 19n/20).

As the unmatched vertices induce a subgraph with at least 19n/20 vertices and less than n edges, they contain an independent set of size at least n/4.

Let, therefore, J denote a maximum independent set of unmatched vertices of G. We have |J| ≥ n/4.

Let K be the remaining unmatched vertices of G.

The third stage consists of matching the vertices of K one by one.

Details similar to those of Stage 2.

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Stage IV.

It remains to match the vertices of J to the remaining unmatched vertices of H, denoting the latter by Q.

Construct a bipartite graph P whose sides are J and Q.

Recall that |J|=|Q| ≥ n/4.

We place an edge from v J to q Q if matching v to q is allowed.

By this we mean that mapping v to q will not violate the packing property.

At the beginning of Stage 4, for each v J , there are at least 19n/20 vertices of H that are non-neighbors of all vertices that are matches of matched neighbors of v. So, the degree of v in P is at least 19n/20-(n-|J|) > |J|/2.

It is not difficult to also show that the degree of each q Q is much larger than |J|/2.

It follows by Hall's Theorem that P has a perfect matching, completing the matching f.14

Concluding remarks

The extremal graph in Ore's Theorem is unique (for all n>5).It is Kn-K1,n-2 .

This is not the case in our more general Theorem 1:

Let H be a graph in which all vertices but one have degree at least 3, and one vertex v is of degree 2 and its two neighbors x and y are adjacent.

By Theorem 1,

One extremal graph is Kn-K1,n-2.

Another extremal graph: The graph T obtained from Kn by deleting a vertex-disjoint union of a star with n-3 edges and a single edge.

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Concluding remarks

all our counter-examples to the [GPW – 2012] conjecture regarding the extremal numbers ex(n,H) for hypergraphs H are based on a local obstruction.

It seems interesting to decide if these are all the possible examples:

Problem:

Is it true that for any k ≥ 2 and any Δ > 0 there is an f = f(Δ) so that for any k-graph H on n vertices and with maximum degree at most Δ, any k-graph on n vertices which contains no copy of H and with ex(n,H) edges, must contain a complete k-graph on at least n-f vertices?

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Thanks

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