Bipartite decomposition of random graphs

For a graph G = ( V , E ) , let ? ( 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, ? ( G ) ? n - α ( G ) , where α ( G ) is the maximum size of an independent set of G. Erd?s conjectured in the 80s that for almost every graph G equality holds, i.e., that for the random graph G ( n , 0.5 ) , ? ( 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 ) , ? ( G ) ? n - α ( G ) - 1 with high probability, and that for some sequences of values of n tending to infinity ? ( G ) ? n - α ( G ) - 2 with probability bounded away from 0. We also study the typical value of ? ( 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 ) , ? ( G ) = n - ? ( α ( G ) ) with high probability.