More on the Bipartite Decomposition of Random Graphs

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.