May 7 th, 2006 On the distribution of edges in random regular graphs Sonny Ben-Shimon and Michael Krivelevich

May 7 th, 2006On the distribution of edges in random regular graphs 2 G(n,p) – probability space on all labeled graphs on n vertices ([n]) each edge chosen with prob. p indep. of others G n,d (dn even) - uniform probability space of all d- regular graphs on n vertices Introduction

May 7 th, 2006On the distribution of edges in random regular graphs 3 how are edges distributed in G(n,p)? how are the edges distributed in G n,d ? this natural question does not have a “trivial” answer pitfalls: all edges are dependent not a product probability space no “natural” generation process Introduction

May 7 th, 2006On the distribution of edges in random regular graphs 4 Introduction applications of analysis bounding  G n,d )=max{| 1 (G n,d )|,| n (G n,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  (G n,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

May 7 th, 2006On 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. G n,d is -jumbled all disjoint pairs of subsets of vertices, U and W, satisfy:

May 7 th, 2006On the distribution of edges in random regular graphs 6 Our results (contd.) Corollary of [ BL04] and Thm 1 : Thm 2 : For w.h.p.  G n,d )= Thm 3 : For d=o(n 1/5 ) and every constant  there exists an integer t=t(n,d,  for which improves result of [AM04] who prove the claim for d=n 1/9-δ for all δ>0 after “correction” of their proof

May 7 th, 2006On the distribution of edges in random regular graphs The configuration model P n,d P G(P) 12d= n=6 dn elements noted by (m,r) s.t P n,d – uniform prob. space on the (dn)!! pairings each pairing corresponds to a d-regular multigraph

May 7 th, 2006On the distribution of edges in random regular graphs 8 The configuration model P n,d all d-regular (simple) graphs are equiprobable each corresponds to pairings define the Simple event in P n,d B event in P n,d and A event in G n,d s.t Thm [MW91] for

May 7 th, 2006On the distribution of edges in random regular graphs 9 Martingale of P n,d P – a pairing in P n,d X – a rand. var. defined on P n,d P(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

May 7 th, 2006On the distribution of edges in random regular graphs 10 Martingale of P n,d (contd.) Thm: if X is a rand. var. on P n,d s.t. whenever P and P’ differ by a simple switch then for all Cor: if Y is a rand. var. on G n,d s.t. Y(G(P))=X(P) for all where X satisfies the conditions of the prev. thm. then

May 7 th, 2006On the distribution of edges in random regular graphs 11 Switchings Q – an integer valued graph parameter Q k – the subset of all graphs from G n,d satisfying Q(G)=k we bound the ratio | Q k |/| Q k+1 | as follows: define a bipartite graph if G can be derived from G’ by a switch G’ Q(G’)=k+1Q(G)=k G

May 7 th, 2006On 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

May 7 th, 2006On 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 G n,d with d=o(n 1/5 ) 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. for every v, N G (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

May 7 th, 2006On 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 G n,d that denotes the minimal size of a set of vertices S for which Lem: s.t. for every n>n 0 where follows the same ideas as [Ł91], [AK97], [AM04] we will also need the following Thm: [FŁ92] for for any w.h.p

May 7 th, 2006On 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-U 0 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)

May 7 th, 2006On 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 P n,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

May 7 th, 2006On the distribution of edges in random regular graphs 17 Thank you for your time

May 7 th, 2006On the distribution of edges in random regular graphs 18 Proof of Thm 1– class I and II (trivial) these are the trivial cases class I class II

May 7 th, 2006On the distribution of edges in random regular graphs 19 class III fix D i - the set of d-regular graphs on n vertices with i edges with one endpoint in U and the other in W Proof of Thm 1– class III (switchings) U W a b x y G at most waysat least ways

May 7 th, 2006On the distribution of edges in random regular graphs 20 Proof of Thm 1– class III (contd.) by assuming (U,W) to be of class III and d=o(n) we have |Di| is monotonically decreasing for set for the chosen values the lower tail is trivial for every j>k Pr[e(U,W)=k]> Pr[e(U,W)=j]

May 7 th, 2006On the distribution of edges in random regular graphs 21 Proof of Thm 1– class III (contd.) summing over all j>k and the union bound on all pairs of class III completes the proof

May 7 th, 2006On the distribution of edges in random regular graphs 22 Proof of Thm 1– class IV (conf. model) class IV straightforward analysis of the Configuration Model - T I is the set of d|I| elements from rows in I let and be the rand. var. in P n,d that count the pairs spanned by T I and between T I and T J resp. let be the set of al pairings in P n,d with k pairs spanned between T I and T J and i pairs spanned by T I. define - it is mono. increasing from 0 to and mono. decreasing from onward.

May 7 th, 2006On the distribution of edges in random regular graphs 23 Proof of Thm 1– class IV (contd.) set set. we prove

May 7 th, 2006On the distribution of edges in random regular graphs 24 Proof of Thm 1– class IV (contd.) summing over all j>k, conditioning on the event Simple and the union bound on all pairs of class IV completes the proof

May 7 th, 2006On the distribution of edges in random regular graphs 25 Proof of Thm 1– class V (martingales) class V martingale concentration result on P n,d fix W={1,…,w} U={s+1,…,w+u} and P a paring P(m) – the subset of pairs with at least one endpoint in one of the first m elements (assuming lexicographic order) note that is a martingale that satisfies

May 7 th, 2006On the distribution of edges in random regular graphs 26 Proof of Thm 1 – class V (contd.) Azuma – Hoeffding and setting union bound on all pairs of class V and conditioning on the event Simple and the completes the proof

May 7 th, 2006On the distribution of edges in random regular graphs 27 Proof of Thm 3 – switchings analysis of edges spanned by subsets using switchings fix C i - the set of d-regular graphs on n vertices with i edges spanned by U U a b x y G at most ways at least ways

May 7 th, 2006On the distribution of edges in random regular graphs 28 Proof of Thm 3 – switchings (contd.) by assuming |U|,d=o(n) we have |Ci| is monotonically decreasing for set we bound the prob. that e(U) surpasses k

May 7 th, 2006On the distribution of edges in random regular graphs 29 Proof of Thm 3 – prep. (contd.) Proof: define the appropriate rand. var. on P n,d : Y’ – the minimal size of S for which if P and P’ differ by a simple switch then define for every note that is a martingale where

May 7 th, 2006On the distribution of edges in random regular graphs 30 Proof of Thm 3 – prep. (contd.) use a martingale concentration result (“pair-Lipschitz”) on P n,d as a corollary we have by setting note moreover, where is a constant

May 7 th, 2006On the distribution of edges in random regular graphs 31 Proof of Thm 3 (contd.) Proof: we find a set U of size s.t. start with U 0 and add sequentially all vertices that violate the condition. Property assures U is of the right size G[U] is 9-degenerate and thus 10-colorable f properly colors G-U by {1,…,t} denote U={u 1,…u k } build an auxiliary graph H as follows…

May 7 th, 2006On the distribution of edges in random regular graphs 32 Proof of Thm 3 (contd.) every vertex corresponds to at most 50 new vertices thus every edge to at most 2500 edges every ind. set in H corresponds to an ind. set in G f induces a proper t-coloring f’ on H U G N G (u k ) N G (u 2 ) N G (u 1 ) ukuk u2u2 u1u1 W1W1 W2W2 WkWk

May 7 th, 2006On the distribution of edges in random regular graphs 33 Proof of Thm 3 (contd.) every union of s sets of has vertices and by spans at most edges for every subset of s such sets there exists one connected by at most edges to the rest assume the vertices of U are ordered in a way that their corresponding neighbor sets are connected by at most edges to the previous ones sequentially, u.a.r choose a color-list J i of 14 colors from the available colors from {1,…,t} a color is available at stage i if there is no edge connecting a vertex in W i colored by it to a vertex in a previous W i’ colored by a previously chosen color

May 7 th, 2006On the distribution of edges in random regular graphs 34 Proof of Thm 3 (contd.) set p i – the prob. that for some there were less than t/2 colors available for J i’ p 1 =0 there are O(1) edges between W i and W i’ for every i’<i so each choice of color for J i’ can make O(1) colors unavailable for J i

May 7 th, 2006On the distribution of edges in random regular graphs 35 Proof of Thm 3 (contd.) there exists a family of color lists for which the corresponding color classes from different lists do not span edges between them from each color list J i we can delete one color class incident with each edge in W i resulting in I i {I i } is a family of color lists of length at least 10. G[U] can be colored from these lists