TY - JOUR

T1 - Bipartite decomposition of random graphs

AU - Alon, Noga

N1 - Publisher Copyright:
© 2015 Elsevier Inc.

PY - 2015/7/1

Y1 - 2015/7/1

N2 - For a graph G=. (. V, E), let τ(. G) denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of G so that each edge of G belongs to exactly one of them. It is easy to see that for every graph G, τ(. G). ≤. n-. α(. G), where α(. G) is the maximum size of an independent set of G. Erdos conjectured in the 80s that for almost every graph G equality holds, i.e., that for the random graph G(. n, 0.5), τ(. G). =. n-. α(. G) with high probability, that is, with probability that tends to 1 as n tends to infinity. Here we show that this conjecture is (slightly) false, proving that for all n in a subset of density 1 in the integers and for G=. G(. n, 0.5), τ(. G). ≤. n-. α(. G). -. 1 with high probability, and that for some sequences of values of n tending to infinity τ(. G). ≤. n-. α(. G). -. 2 with probability bounded away from 0. We also study the typical value of τ(. G) for random graphs G=. G(. n, p) with p<. 0.5 and show that there is an absolute positive constant c so that for all p≤. c and for G=. G(. n, p), τ(. G). =. n-. Θ(α(. G)) with high probability.

AB - For a graph G=. (. V, E), let τ(. G) denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of G so that each edge of G belongs to exactly one of them. It is easy to see that for every graph G, τ(. G). ≤. n-. α(. G), where α(. G) is the maximum size of an independent set of G. Erdos conjectured in the 80s that for almost every graph G equality holds, i.e., that for the random graph G(. n, 0.5), τ(. G). =. n-. α(. G) with high probability, that is, with probability that tends to 1 as n tends to infinity. Here we show that this conjecture is (slightly) false, proving that for all n in a subset of density 1 in the integers and for G=. G(. n, 0.5), τ(. G). ≤. n-. α(. G). -. 1 with high probability, and that for some sequences of values of n tending to infinity τ(. G). ≤. n-. α(. G). -. 2 with probability bounded away from 0. We also study the typical value of τ(. G) for random graphs G=. G(. n, p) with p<. 0.5 and show that there is an absolute positive constant c so that for all p≤. c and for G=. G(. n, p), τ(. G). =. n-. Θ(α(. G)) with high probability.

KW - Bipartite decomposition

KW - Random graphs

KW - Stein-Chen method

UR - http://www.scopus.com/inward/record.url?scp=84929050312&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2015.03.001

DO - 10.1016/j.jctb.2015.03.001

M3 - מאמר

AN - SCOPUS:84929050312

VL - 113

SP - 220

EP - 235

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

ER -