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Universality, Tolerance, Chaos and Order

Noga Alon, Tel Aviv University

Szemerédi’s ConferenceBudapest, August 2010

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Universality and Tolerance

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Universal Graphs

Definition: H- A family of graphs. G is H- universal if it contains every member

of H as a subgraph

Example: G=

is H-universal for the family of all 2-regular graphs on 7 vertices

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Objective: Construct sparse H-universal graphsfor interesting families H

Motivation: VLSI circuit design

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Some early results:

Sparse universal graphs for forestsChung, Graham and Pippenger (78)Friedman and Pippenger (87)

Sparse universal graphs for planar graphs and graphs with small separatorsBondy (71)Rödl (81)Babai,Chung,Erdős and Graham (82)Capalbo (99)Capalbo and Kosaraju (99)

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In these cases the number of edges is linear or nearly linear

This is not the case for general cubic graphs

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Universal graphs for bounded-degree graphs:H(k,n)=all graphs on n vertices, max-degree ≤ kQuestion: Estimate the minimum possible number of edges of an H(k,n)-universal graph, and of one

on n vertices

A,Capalbo,Kohayakawa,Rödl,Ruciński,Szemerédi:Universality and Tolerance (00):

Ω(n2-2/k) edges are needed, O(n2-1/k log1/k n) suffice(using more than n vertices)O(n2-c/k log k) edges suffice (with n vertices)

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Moreover: O(n2-1/2k log1/2k n) edges suffice for a fault tolerant universal graph: every set of 1% of its edges contains all bipartite graphs on n+n vertices with maximum degree at most k

ACKRRS (01): O(n2-2/k log 1+8/k n) edges suffice (using more than n vertices)

A and Asodi (02): For k=3, n vertices, O(n1.87..) edges suffice

Dellamonica,Kohayakawa,Rödl and Ruciński (08): n vertices, O(n2-1/2k log1/k n) edges suffice

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A+ Capalbo (08):

Theorem 1: For all k ≥3 there is c=c(k) and an explicit H(k,n)-universal G on n vertices with at most c n2-2/k log4/k n edges.

Theorem 2: For all k ≥ 3 there is c=c(k) and an explicit H(k,n)-universal G with at most c n2-2/k edges.

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The proof of Theorem 1 is probabilistic, based on therapid mixing of random walks on expanders.

The proof of Theorem 2 applies properties of high-girthexpanders, and provides a deterministic embedding procedure.

The proof for even k is simpler, the one for odd k requires an additional effort: a new graph decomposition result.

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Theorem 2 for k=4: The minimum possible number of edges of a graph that contains a copy of every graph on n vertices with maximum degree at most 4 is Θ(n3/2)

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The lower bound:Simple counting: there are “many” 4-regular graphs on n vertices, and a graph with m edges cannot contain too many subgraphs with 2n edges

The upper bound:Construction using high-girth expanders

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The construction:Let a,d be absolute constants, put m=a n1/2, andlet F be a d-regular Ramanujan expander of girth atleast ⅔ logd-1m. Thus all nontrivial eigenvaluesof F are of absolute value at most 2(d-1)1/2.

Define G=(V,E), where V=(V(F))2 and (a1,a2) isadjacent to (b1,b2) iff ai and bi are within distance 2 in F for i=1 and/or i=2.

Clearly |E|=O(n3/2).

Main claim: G is H(4,n)-universal.

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A homomorphism from a graph Z to a graph T is amapping of V(Z) to V(T) such that adjacent verticesin Z are mapped to adjacent vertices in T.

Thus there is an injective homomorphism from Z toT iff Z is a subgraph of T.

Pn - the path of length n. A homomorphism from Pn

to F is a walk on F.

The k-th power Tk of a graph T is the graph on V(T)in which two vertices are adjacent iff their distancein T is at most k.

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Let H be a graph on n vertices with maximum degreeat most 4.

By Petersen’s Theorem H can be decomposed intotwo spanning subgraphs H1,H2, each having max.degree at most 2.

There are bijective homomorphisms gi from Hi to Pn2,

To embed H in G we define homomorphisms fi from Pn

to F so that f(v)=(f1(g1(v)), f2(g2(v)) ) is an injectivehomomorphism from H to G.

This is done by defining each fi as an appropriate non-back-tracking walk on F.

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The existence of the required walks fi is proved using the spectral properties of the expanderF and the fact it has high girth.

The construction of universal graphs for H(k,n) with k>4 even is similar

The odd case requires more efforts

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A graph is thin if every connected component of it iseither a subgraph of a cycle with pendant edges

or a graph with max. degree 3 and at most two vertices of degree 3

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Fact: Every thin graph can be mapped homomorphically and bijectively to the forth power of a path.

Theorem: Let H be a graph of maximum degree k.Then there are k thin spanning subgraphs H1, H2, … ,Hk of H, so that each edge of H lies in twoof the graphs Hi.

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A universal graph for H(k,n):

Let F be a high-girth Ramanujan graph on m=a n1/k vertices.

Construct G=(V(G),E(G)) as follows:V(G)=(V(F))k

(a1,a2, … ,ak) and (b1,b2, … ,bk) are adjacent iffthere are at least two indices i so that ai and bi

are within distance 4 in F.

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Open:

Is there an H(k,n)-universal graph on n verticeswith O(n2—2/k) edges ?

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Motzkin: The essence of Ramsey Theory is that complete chaos is impossible: every sufficiently large system contains a substantial ordered one.

Chaos and Order

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Chvátal, Rödl, Szemerédi and Trotter (83):The Ramsey number of any bounded degree graphis linear in its number of vertices, that is:

for any fixed k, there is a constant c=c(k) so thatfor any graph G with maximum degree k and n vertices, any red-blue coloring of the edges of the complete graph on c n vertices contains a monochromaticcopy of G.

The proof is based on Szemerédi’s Regularity Lemma

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Kohayakawa, Rödl, Schacht and Szemerédi (2010):The complete graph can be replaced by a sparser(random) graph, with only O(n2-1/k log1/k n) edges.

Moreover: any red-blue coloring of such a graph contains a monchromatic H(k,n)-universal graph.

The proof combines the regularity method with a subtle embedding lemma

Open: is there such a graph with only O(n2-2/k) edges ?

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Subgraph containment problems

A and Marx (2010+): some of the techniques in the investigation of universal graphs for bounded degree graphs are relevant to the study of the complexity of subgraph containment problems

The colored H-subgraph problem: given a fixed graph H on the vertices {1,2, … , h}, decide if an input graph G with n vertices colored by 1,2,..,hcontains a copy of H respecting the coloring.

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Example H=

G=

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Freuder (90), A,Yuster and Zwick (95): It canbe also solved in time 2O(h) nO(w) , where w is thetree-width of H.

Clearly: The H-colored subgraph problem can be solved in time O(nh)

Marx (07): Assuming the exponential time hypothesis, it cannot be solved in time no(w/ log w)

The exponential time hypothesis [ Impagliazzo, Paturi and Zane (01) ]:Satisfiability on m variables cannot be solvedin time 2o(m).

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A rough sketch of the proof of Marx:

Represent the satisfiability formula by a graph F with O(m) edges

Call a function mapping each vertex of F to a connected subset of H an embedding of depth d (of F into H) if the endpoints of each edge of F are mapped to sets that arewithin distance 1 in H, and the inverse image of each vertex of H is of size at most d

Show, crucially, that if the tree-width of H is w, then there is an embedding of depth at most O( m log w /w ) (of F into H).

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Construct a graph G by replacing each vertex i of Hby a set of vertices, colored i, corresponding to all2O( m log w/ w) possible assignments of the variables mapped to i, and define the edges of G to ensure that satisfying assignments will correspond to copies of H.

If we can now solve the colored H–subgraph problem in time no( w / log w ), we’ll solve satisfiability in time [2O(m log w / w) ]o(w / log w) = 2o(m), contradicting the exponential time hypothesis. ■

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It thus follows that to improve the hardness result to a tight no(w) bound, it will suffice to show thatany graph F with O(m) edges admits an embedding of depth at most O( m /w) into any graph of tree width w.

A and Marx (2010+): This is false, namely, thelog w extra term cannot be omitted.

The proof is by showing that certain balanced homomorphisms of random cubic graphs into expanders do not exist. The study of those appliessimilar ideas to those used in the investigation of universal graphs.

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This is meant to stay somewhat obscureHappy Birthday, Endre !