The Effect of Induced Subgraphs on Quasi-randomness

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

2. Background 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. 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.

3. 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: • Any set of vertices U  V spans  ½p|U|2 edges • Any set of vertices U  V of size ½n spans  ½p|U|2 edges • 1(G) pn and 2(G) = o(pn) • For any graph H on h vertices, G hasnhpe(H) copies of H • G contains  ½pn2edges and p-4n4 copies of C4 • All but o(n2) of the pairs u,v have p2n common neighbors “” = (1+o(1)) Note: All the above hold whp in G(n,p). Definition: A graph that satisfies any (and therefore all) the above properties is p-quasi-random, or just quasi-random.

4. Background Relation to (theoretical) computer science: 1. Conditions of randomness that are verifiable in polynomial time. For example, using number of C4, or using 2(G). 2. Algorithmic version of Szemeredi’s regularity-lemma [ADLRY ‘95], uses equivalence between regularity and number of C4. Relation to Extremal Combinatorics: 1. Central in the stronghypergraph generalizations of Szemeredi’s regularity-lemma.

5. 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? No! Any (natural) property that holds in G(n,p)whp? Example:Having  ½pn2 edges and p-3n3 copies of K3 is not a quasi-random property. Example:Having  ¼ pn2 edges crossing all cuts of size (½n,½n)is not a quasi-random property. Recall that if we replace K3 with C4 we do get a quasi-random property. …but if we consider cuts of size (¼n,¾n) then the property is quasi-random.

6. Subgraph and Quasi-randomness The effect of subgraphs on quasi-randomness: 1. Having the correct number of edges + correct number of C4 is a quasi-random property. True also for any C2k. 2. Having the correct number of edges + correct number of K3 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|hpe(H) copies of H, then G is p-quasi-random. Intutition:“Randomness is a hereditary property”.

7. Induced Subgraph and Quasi-Rand [Simonovits and Sos ’97]:For any H, if all vertex sets U  V span |U|hpe(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? Related Question:Can we expect to be able to deduce from the distribution of a single H that a graph is p-quasi-random. 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.

8. 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 ’97]:For any H, if all vertex sets U  V span |U|hpe(H) copies of H, then G is p-quasi-random. Definition: 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? [Simonovits and Sos ’03]:No. There are non quasi-random graphs, where all U  V span |U|3p2(1-p) induced copies of P3 .

9. A New Formulation 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 • Lemma:Fix any H on h vertices. The following are equivalent: • G is such that all UV span |U|hH(p) induced copies of H. • G is such that all h-tupleU1,…,Uh of (arbitrary) size m span •  h!mh H(p) induced copies of H with one vertex in each Ui . Observation:Consider any orderedh-tupleU1,…,Uh of (arbitrary) size m in G(n,p). We actually expect U1,…,Uh to span  mh H(p) induced copies of H with the vertex in Ui playing the role of vertex vi of H. Definition: In that case, we say that U1,…,Uh have the correct number of induced embedded copies of H.

10. 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! • Assume all ordered h-tuplesU1,…,Uh 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 U1,…,Uh .

11. 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 V1,…,Vk, and most of the bipartite graphs on (Vi,Vj) 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.

12. Proof Overview V1 V2 V4 V3 Fact:If G is an h-partite graph on V1,…,Vh and all the bipartite graphs (Vi,Vj) are quasi-random, then the number of induced copies of H is determined by the densities between (Vi,Vj). In fact even the number of induced embedded copies of H.  x1,2 · x1,3 ·x1,4 · x2,3 · x2,4 · (1-x2,3) The density of is density = x2,4 denstiy = x1,3

13. Proof Overview [Szemeredi’s regularity lemma ‘79]:Any graph has a k-partition into V1,…,Vk s.t. most graphs on (Vi,Vj)are quasi- random. But not necessarily with the same density… We will show that in such a partition most densities are p or q. Fact:Given a partition where most bipartite graphs (Vi,Vj) are quasi-random, we “know” the number of induced copies of H. Assumption on G:We “know” the number of induced embedded copies of H in each h-tuple of vertex sets V1…,Vh. We get (k)h polynomial equations relating the densities (Vi,Vj). We will show that the only solution is p or q.

14. 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. 1. w(i,j) stands for the density between Vi and Vj. 2.W()(p)due to our assumption on G. 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.

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

16. Proof Overview Proof idea:Suppose that for all :[h][k] we have First step:Show that the only solution is xij {p,q}. 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. k-element sets AS,T = 1 iif ST [Gottlieb ‘66]:If r  k+h, then rank(A(r,h,k)) = . h-element sets

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

18. Thank You