On the distribution of edges in random regular graphs

1. On the distribution of edges in random regular graphs Sonny Ben-Shimon and Michael Krivelevich On the distribution of edges in random regular graphs

2. Introduction • G(n,p) – probability space on all labeled graphs on n vertices ([n]) • each edge chosen with prob. p indep. of others • Gn,d (dn even) - uniform probability space of all d-regular graphs on n vertices On the distribution of edges in random regular graphs

3. Introduction • how are edges distributed in G(n,p)? • how are the edges distributed in Gn,d? • this natural question does not have a “trivial” answer • pitfalls: • all edges are dependent • not a product probability space • no “natural” generation process On the distribution of edges in random regular graphs

4. Introduction • applications of analysis • bounding (Gn,d)=max{|1(Gn,d)|,|n(Gn,d)|} based on:Thm:[BL04] Let G be a d-regular graph on n vertices. If all disjoint pairs of subsets of vertices, U and W, satisfy:then (G)=O((1+log(d/)) • 2-point concentration of c(Gn,d) based on result of [AK97] on the G(n,p) model • proof consists of showing that sets of various cardinalities do not span “too many” edges On the distribution of edges in random regular graphs

5. Our results • Defn: A d-regular graph on n vertices is -jumbled, if for every two disjoint subsets of vertices, U and W, • Thm 1: W.h.p. Gn,d is -jumbled • all disjoint pairs of subsets of vertices, U and W, satisfy: On the distribution of edges in random regular graphs

6. Our results (contd.) • Corollary of [BL04] and Thm 1:Thm 2: For w.h.p. (Gn,d)= • Thm 3: For d=o(n1/5) and every constant >0 there exists an integer t=t(n,d, ) for which • improves result of [AM04] who prove the claim for d=n1/9-δ for all δ>0 • after “correction” of their proof On the distribution of edges in random regular graphs

7. The configuration model Pn,d • dn elements noted by (m,r) s.t • Pn,d – uniform prob. space on the (dn)!! pairings • each pairing corresponds to a d-regular multigraph 2 1 2 d=3 1 1 3 2 3 4 6 4 5 n=6 5 G(P) P On the distribution of edges in random regular graphs

8. The configuration model Pn,d • all d-regular (simple) graphs are equiprobable • each corresponds to pairings • define the Simple event in Pn,dB event in Pn,d and A event in Gn,d s.t • Thm [MW91] for On the distribution of edges in random regular graphs

9. Martingale of Pn,d • P – a pairing in Pn,dX – a rand. var. defined on Pn,dP(m) – the subset of pairs with at least one endpoint in one of the first m elements (assuming lexicographic order) • the “pair exposure” martingale, • analogue of the “edge exposure” martingale for random graphs • the Azuma-Hoeffding concentration result can be applies On the distribution of edges in random regular graphs

10. Martingale of Pn,d (contd.) • Thm: if X is a rand. var. on Pn,d s.t.whenever P and P’ differ by a simple switch then for all • Cor: if Y is a rand. var. on Gn,d s.t. Y(G(P))=X(P) forall where X satisfies the conditions of the prev. thm. then On the distribution of edges in random regular graphs

11. Switchings • Q – an integer valued graph parameterQk – the subset of all graphs from Gn,d satisfying Q(G)=k • we bound the ratio | Qk |/| Qk+1 | as follows: • define a bipartite graph if G can be derived from G’ by a switch Q(G’)=k+1 Q(G)=k G G’ On the distribution of edges in random regular graphs

12. Proof of Thm 1 • Proof: Classify all pairs (U,W) • class I • class II • class III • class IV • class V On the distribution of edges in random regular graphs

13. Proof of Thm 3 – prep. (edge dist.) • using switchings and union bound we prove some results on the distribution of edges in Gn,d with d=o(n1/5) • property - w.h.p. every subset of vertices spans at most 5u edges • property - w.h.p. every subset of vertices spans at most edges • property - w.h.p. every subset of vertices spans at most edges • property - w.h.p. for every v, NG(v) spans at most 4 edges • property - w.h.p. the number of paths of length 3 between any two vertices, u and w, is at most 10 On the distribution of edges in random regular graphs

14. Proof of Thm 3 – prep. (contd.) • for every we define to be the least integer for which • Y(G) – the rand. var in Gn,d that denotes the minimal size of a set of vertices S for which • Lem: s.t. for everyn>n0 where • follows the same ideas as [Ł91],[AK97],[AM04] • we will also need the followingThm: [FŁ92] for for any w.h.p On the distribution of edges in random regular graphs

15. Proof of Thm 3 - main prop. • Thm 3 follows from: • Prop: let G be a d-regular graph on n vertices with all - properties where suppose that and that there exists of at most s.t. G-U0 is t-colorable.G is (t+1)-colorable for large enough values of n. • set then based on the prop. • the case of is covered by [AM04] (after minor correction) On the distribution of edges in random regular graphs

16. Further research • expand the range for which Thm 1 applies (and thus, Thm 2 as well) • requires to eliminate the use of the configuration model • for this requires us to deal with events of Pn,d of very low probability • to eliminate the log factor in Thm 1 – try and give a w.h.p [BL04] lem. rather than deterministic • using and analogue of the vertex-exposure martingale to extend the 2-point concentration of the chromatic number for larger values of d On the distribution of edges in random regular graphs

17. Thank you for your time On the distribution of edges in random regular graphs