may 7 th, 2006 on the distribution of edges in random regular graphs sonny ben-shimon and michael...

May 7th, 2006 On the distribution of edges in random regular graphs

On the distribution of edges in random regular graphs

Sonny Ben-Shimon and Michael

Krivelevich

May 7th, 2006 On 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

Gn,d (dn even) - uniform probability space of all d-

regular graphs on n vertices

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

Introduction

U,W [n]

U

e(U) B ,p2

e(U,W) B U W ,p


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 (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

U W de U,W U W

n


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:

U W d

e U,W U Wn

d,O d

d

d,

U W de U,W O U W d

n


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

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

d o n

n,dPr t (G ) t 1 1

O d logd


1

2

6

3

4

5

The configuration model Pn,d

P G(P)

1 2 d=31

2345

n=6

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

1 m n,1 r d


The configuration model Pn,d

all d-regular (simple) graphs are equiprobable each corresponds to pairings

define the Simple event in Pn,d

B event in Pn,d and A event in Gn,d s.t

Thm [MW91] for

nd!

B Simple AP G(P) B

A B SimpleSimplePr[ ]

Pr[ ] Pr[ | ]Pr[ ]

d o n

Simple2 3 21 d d d

Pr[ ] exp O4 12 n


Martingale of Pn,d

P – a pairing in Pn,d

X – a rand. var. defined on Pn,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

0 dn 1X , ,X

mX (P) E X Q | P(m) Q

0X (P) E X dn 1X (P) X P


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) for

all where X satisfies the conditions of

the prev. thm. then

X P X P' c

0

2 2Pr X E X 2exp / dnc

SimpleP

2 22exp / dncPr Y E Y

Pr Simple


Switchings

Q – an integer valued graph parameter

Qk – 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

k k 1H Q Q ,F

k k 1G Q ,G' Q G,G' F

k k 1

k k1 k H H 2 k 1

G Q G' Q

d Q d (G) F d (G') d Q

k k

k 1 k 1 2Q / Q d / d

j 1

j r k 1 kk r

r j Pr Q j Q / Q Q / Q

G’

Q(G’)=k+1Q(G)=k

G


Proof of Thm 1

U W

UW

d

U W d

Proof: Classify all pairs (U,W) U,W n

class I

class II

class IIIU W d

C U W dn

U

Wd

n

d Uc

class IVU W d

C U W dn

n

Uc

class Vn

Uc


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


spans at most edges


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

3u O nd

9/ 10u O n

5u d

u nlnn/ d25u d/ n3

1

2

4

5


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

every n>n0

where

follows the same ideas as [Ł91],[AK97],[AM04]

we will also need the following

Thm: [FŁ92] for for any

w.h.p

0 1 n,d, n,d

Pr G

G \ S

0 0 0C C ,n n

3

n,dPr Y G C nd d o n

1/ 30d d d 0

n,dG d/ 2lnd


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)

1/ 10 1/ 5n d n

G t d/ 2lnn

0U V G 3c nd

t n,d, / 3

n,d n,dPr G t G t 1 / 3 1 o 1 / 3 1/ 9d n


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

nd n , Pr[Simple] o e


Thank you for your time

may 7 th, 2006 on the distribution of edges in random regular graphs sonny ben-shimon and michael...

Documents