The Effect of Induced Subgraphs on Quasi-randomness Asaf Shapira & Raphael Yuster

Background Chung, Graham, and Wilson ’89, Thomason ‘87: 1. Defined the notion of a p-quasi-random = A graph that “behaves” like a typical graph generated by G(n,p). 2. Proved that several “natural” properties guarantee that a graph is p-quasi-random. Abstract Question: What it means for a graph to be random? “Concrete” problem: Which graph properties “force” a graph to behave like a “truly” random one.

The CGW Theorem Theorem [CGW ‘89]: Fix any 0<p<1, and let G=(V,E) be a graph on n vertices. The following are equivalent: 1.Any set of vertices U  V spans  ½p|U| 2 edges 2.Any set of vertices U  V of size ½n spans  ½p|U| 2 edges 3. 1 (G)  pn and 2 (G) = o(pn) 4.For any graph H on h vertices, G has  n h p e(H) copies of H 5. G contains  ½pn 2 edges and  p -4 n 4 copies of C 4 6.All but o(n 2 ) of the pairs u,v have  p 2 n common neighbors Definition: A graph that satisfies any (and therefore all) the above properties is p-quasi-random, or just quasi-random. “  ” = (1+o(1)) Note: All the above hold whp in G(n,p).

Background Relation to (theoretical) computer science: 1. Conditions of randomness that are verifiable in polynomial time. For example, using number of C 4, or using 2 (G). 2. Algorithmic version of Szemeredi’s regularity-lemma [ADLRY ‘95], uses equivalence between regularity and number of C 4. Relation to Extremal Combinatorics: 1. Central in the strong hypergraph generalizations of Szemeredi’s regularity-lemma.

More on the CGW Theorem Definition: Say that a graph property is quasi-random if it is equivalent to the properties in the CGW theorem. “The” Question: Which graph properties are quasi-random? Any (natural) property that holds in G(n,p) whp? Example: Having  ½pn 2 edges and  p -3 n 3 copies of K 3 is not a quasi-random property. Example: Having  ¼ pn 2 edges crossing all cuts of size (½n,½n) is not a quasi-random property. Recall that if we replace K 3 with C 4 we do get a quasi-random property. No! …but if we consider cuts of size (¼n,¾n) then the property is quasi-random.

Subgraph and Quasi-randomness The effect of subgraphs on quasi-randomness: 1. Having the correct number of edges + correct number of C 4 is a quasi-random property. True also for any C 2k. 2. Having the correct number of edges + correct number of K 3 is not a quasi-random property. True for any non-bipartite H. Question: Is it true that for any single H, the “distribution” of copies of H affects the quasi-randomness of a graph? [Simonovits and Sos ’97]: Yes. If all vertex sets U  V span  |U| h p e(H) copies of H, then G is p-quasi-random. Intutition: “Randomness is a hereditary property”.

Induced Subgraph and Quasi-Rand Related Question: Can we expect to be able to deduce from the distribution of a single H that a graph is p-quasi-random. [Simonovits and Sos ’97]: For any H, if all vertex sets U  V span  |U| h p e(H) copies of H, then G is p-quasi-random. Question: Is it true that for any single H, the “distribution” of induced copies of H affects the quasi-randomness of a graph? Answer: No. For any p and H, there is a q=q(p,H) for which G(n,p) and G(n,q) behave identically w.r.t. induced copies of H.

Induced Subgraph and Quasi-Rand Question: Is it true that for any single H, the “distribution” of induced copies of H affects the quasi-randomness of a graph? [Simonovits and Sos ’03]: No. There are non quasi-random graphs, where all U  V span  |U| 3 p 2 (1-p) induced copies of P 3. [Simonovits and Sos ’97]: For any H, if all vertex sets U  V span  |U| h p e(H) copies of H, then G is p-quasi-random. Question: Is it true that for any H, if all vertex sets U  V span  |U| h  H (p) induced copies of H, then G is quasi-random? Definition:

A New Formulation Lemma: Fix any H on h vertices. The following are equivalent: 1. G is such that all U  V span  |U| h  H (p) induced copies of H. 2. G is such that all h-tuple U 1,…,U h of (arbitrary) size m span  h!m h  H (p) induced copies of H with one vertex in each U i. Observation: Consider any ordered h-tuple U 1,…,U h of (arbitrary) size m in G(n,p). We actually expect U 1,…,U h to span  m h  H (p) induced copies of H with the vertex in U i playing the role of vertex v i of H. Question: Is it true that for any H, if all vertex sets U  V span  |U| h  H (p) induced copies of H, then G is quasi-random? NO Definition: In that case, we say that U 1,…,U h have the correct number of induced embedded copies of H.

Main Result Question: Is it true that for any single H, the “distribution” of induced copies of H affects the quasi-randomness of a graph? Theorem [S-Yuster ‘07]: Yes! 1.Assume all ordered h-tuples U 1,…,U h of (arbitrary) size m span the correct number of induced embedded copies of H. Then G is quasi-random. 2. In fact, G is either p-quasi-random or q-quasi-random. Note: 1. We can’t expect to show that G is p-quasi-random. 2. We can’t consider only the number of induced copies of H in U 1,…,U h.

Proof Overview Lemma [SS 91]: If G is composed of quasi-random graphs with the same density, then G itself is quasi-random. That is, Suppose G is a k-partite graph on vertex sets V 1,…,V k, and most of the bipartite graphs on (V i,V j ) are p-quasi-random. Then G is also p-quasi-random. Overall strategy: Show that G has a k-partition, where most of the quasi-random bipartite graphs have density p or most have density q.

Proof Overview Fact: If G is an h-partite graph on V 1,…,V h and all the bipartite graphs (V i,V j ) are quasi-random, then the number of induced copies of H is determined by the densities between (V i,V j ). denstiy = x 1,3 density = x 2,4  x 1,2 · x 1,3 · x 1,4 · x 2,3 · x 2,4 · (1-x 2,3 ) V1V1 V3V3 V2V2 V4V4 The density of is In fact even the number of induced embedded copies of H.

Proof Overview We will show that in such a partition most densities are p or q. [Szemeredi’s regularity lemma ‘79]: Any graph has a k-partition into V 1,…,V k s.t. most graphs on (V i,V j ) are quasi- random. But not necessarily with the same density… Assumption on G: We “know” the number of induced embedded copies of H in each h-tuple of vertex sets V 1 …,V h. Fact: Given a partition where most bipartite graphs (V i,V j ) are quasi-random, we “know” the number of induced copies of H. We get (k) h polynomial equations relating the densities (V i,V j ). We will show that the only solution is p or q.

Proof Overview Let W be a weighted complete graph on k vertices, with weights 0  w(i,j)  1. For every mapping  :[h]  [k] define Key Lemma: Suppose that for all  :[h]  [k] we have W(  )  (p). Then either most w(i,j)  p or most w(i,j)  q. Notes: We cannot expect to show that most w(i,j)  p. Also, it is NOT true that either all w(i,j)  p or all w(i,j)  q. 1. w(i,j) stands for the density between V i and V j. 2. W(  )   (p) due to our assumption on G.

Proof Overview Key Lemma: Suppose that for all  :[h]  [k] we have Proof idea: Introduce unknowns x ij for each w(i,j). Consider the system of polynomial equations: Then either most w(i,j)  p or most w(i,j)  q. First step: Show that the only solution is x ij  {p,q}.

Proof Overview Proof idea: Suppose that for all  :[h]  [k] we have First step: Show that the only solution is x ij  {p,q}. [Gottlieb ‘66]: If r  k+h, then rank(A(r,h,k)) =. Definition: For integers k  h  r, let A(r,h,k) be the inclusion matrix of the k-element subsets of [r] and its h-element subsets. h-element sets k-element sets A S,T = 1 iif S  T

Proof Overview 3)Use Regularity-lemma + Packing results ( Rodl or Wilson ) to conclude that G is composed of quasi-random graphs with the same densities. 2)Some more (non-trivial) arguments needed to show that in fact most are p or most are q. 1)After (appropriate) manipulations this gives that the unique solution uses p and q.

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