Bipartite decomposition of random graphs

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.