The tur n number of sparse spanning graphs
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The Turán number of sparse spanning graphs. Raphael Yuster joint work with Noga Alon. Banff 2012. [ Ore – 1961 ] A non-Hamiltonian graph of order n has at most edges. ex( n,H ) is the maximum number of edges in a graph of order n not containing a subgraph isomorphic to H .

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The Turán number of sparse spanning graphs

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The tur n number of sparse spanning graphs

The Turánnumber of sparse spanning graphs

Raphael Yuster

joint work with

Noga Alon

Banff 2012


The tur n number of sparse spanning graphs

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

  • ex(n,H)is the maximum number of edges in a graph of order nnot 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 nsuff.large).It is of the form where P is a specific fixed (k-1)-graph.

t


The tur n number of sparse spanning graphs

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

Suppose that His a k-graph of order nand with, say, bounded max degree.

It is natural to suspect that for n sufficiently large,where L is aset of (k-1)-graphs that depending on neighborhoods in H.

A conjecture raised in [GPW – 2012] asserts that it suffices to take Lto be the set of linksof 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).


The tur n number of sparse spanning graphs

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 His a graph of order nwith mindegδ>0and bounded maxdeg, then

assuming nis 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 nsufficiently large, if H is any graph of order n with no isolated vertices and , then


The tur n number of sparse spanning graphs

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

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

    • Clearly, Δ(H)=(n-1)/kand δ(H)=δ.

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

    • Hence, if Gis Kn - Kk+2, then His not a spanning subgraphof G.

    • However, Ghas more than edges.


The tur n number of sparse spanning graphs

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

Proposition 2:Let sbe a large integer, n=1+5sand let V1… Vs{x}where |Vi|=5.

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

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

Proof: Take T to be U1U2  U3{x,y}where |Ui|=(n-2)/3.The edges are all the triples of U1U2U3 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.


The tur n number of sparse spanning graphs

Proof preliminaries

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

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

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

It suffices to prove that G and Hpack.

Equivalently, a bijectionf : V  Wsuch 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/2d(vi) ≤ 2n/i


The tur n number of sparse spanning graphs

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

S1 consists of non-neighbors of v1that have small degree (less than 2n1/2)

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

Each Siis chosen with maximum cardinality, under this restriction.

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

vi

Si

N(vi)

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 Ciare relatively small (less than 4n1/2)the first few Diare relatively large (at least0.05n1/2fori=2,…,n1/2)


The tur n number of sparse spanning graphs

Proof outline

The construction of the bijectionf : V  Wis done in four stages.

At each point of the construction, some vertices of Vare matchedto some vertices of Wwhile the other vertices of V and Ware yet unmatched.Initially, all vertices are unmatched.

We always maintain the packingproperty:

for two matched vertices u,vVwith (u,v)Ewe have (f(u), f(v))F.

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


The tur n number of sparse spanning graphs

Stage 1.

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

As N(w)= δand 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+1pairs are matched.

v1

w

|S1|≥ δ

|N(w)|=δ

N(v1)

S1

G

H

Other vertices

N(w)


The tur n number of sparse spanning graphs

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:

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

Any matched vertex of Gother than {v1,…,vi}is contained in some Bjwhere j ≤ i.

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

These invariants clearly hold after stage 1.


The tur n number of sparse spanning graphs

vi

Observe: it is really yet unmatched

Z

X

Y

SiNon-neighbors

G

unmatched neighbors

Other matched neighbors

Bi

matched vjj < i

Di =Bi-j<iBj

Y j<iBjY N(vi), Y Ci|Y|<5n1/2

|X|<i<k<n1/2

Lemma 1

|Di| ≥ n1/2/20, lemma 1

R

Unmatchedneighbors of w

|R|≤≤ n1/2/40

w

non-neighbors of T

The matches of X  Y in H

H

Q

T

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

|Q| ≥ n-|T|

Is there an unmatched vertex in Q?

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


The tur n number of sparse spanning graphs

  • Stage I1I.

  • We are guaranteed that the unmatched vertices of Ghave degree ≤2n1/2.

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

  • As the unmatched vertices induce a subgraph with at least 19n/20 vertices and less than nedges, 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 Kone by one.

  • Details similar to those of Stage 2.


The tur n number of sparse spanning graphs

Stage IV.

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

Construct a bipartite graph Pwhose sides are Jand Q.

Recall that |J|=|Q| ≥ n/4.

We place an edge from v J to q Qif matching vto qis allowed.

By this we mean that mapping vto qwill not violate the packing property.

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

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

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


The tur n number of sparse spanning graphs

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 bea 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 yare adjacent.

  • By Theorem 1,

  • One extremal graph is Kn-K1,n-2.

  • Another extremalgraph: The graph T obtained from Knby deleting a vertex-disjoint union of a star with n-3edges and a single edge.


The tur n number of sparse spanning graphs

Concluding remarks

  • all our counter-examples to the [GPW – 2012] conjecture regarding the extremal numbers ex(n,H)for hypergraphsH 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 > 0there is an f = f(Δ) so that for any k-graph H on nvertices and with maximum degree at most Δ, any k-graph on n vertices which contains no copy of Hand with ex(n,H)edges, must contain a complete k-graph on at least n-fvertices?


The tur n number of sparse spanning graphs

Thanks


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