TY - JOUR

T1 - More on the Bipartite Decomposition of Random Graphs

AU - Alon, Noga

AU - Bohman, Tom

AU - Huang, Hao

N1 - Publisher Copyright:
© 2016 Wiley Periodicals, Inc.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - For a graph G = (V, E), let bp(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, bp(G) ≤ n − α(G), where α(G) is the maximum size of an independent set of G. Erdős conjectured in the 80s that for almost every graph G equality holds, that is that for the random graph G(n, 0.5), bp(G) = n − α(G) with high probability, that is with probability that tends to 1 as n tends to infinity. The first author showed that this is slightly false, proving that for most values of n tending to infinity and for G = G(n, 0.5), bp(G) ≤ n − α(G) − 1 with high probability. We prove a stronger bound: there exists an absolute constant c > 0 so that bp(G) ≤ n − (1 + c)α(G) with high probability.

AB - For a graph G = (V, E), let bp(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, bp(G) ≤ n − α(G), where α(G) is the maximum size of an independent set of G. Erdős conjectured in the 80s that for almost every graph G equality holds, that is that for the random graph G(n, 0.5), bp(G) = n − α(G) with high probability, that is with probability that tends to 1 as n tends to infinity. The first author showed that this is slightly false, proving that for most values of n tending to infinity and for G = G(n, 0.5), bp(G) ≤ n − α(G) − 1 with high probability. We prove a stronger bound: there exists an absolute constant c > 0 so that bp(G) ≤ n − (1 + c)α(G) with high probability.

KW - bipartite decomposition

KW - random graph

UR - http://www.scopus.com/inward/record.url?scp=84959200524&partnerID=8YFLogxK

U2 - 10.1002/jgt.22010

DO - 10.1002/jgt.22010

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AN - SCOPUS:84959200524

SN - 0364-9024

VL - 84

SP - 45

EP - 52

JO - Journal of Graph Theory

JF - Journal of Graph Theory

IS - 1

ER -