More on the bipartite decomposition of random graphs

For a graph $G=(V,E)$, let $bc(G)$ 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 $G$, $bc(G) \leq n -\alpha(G)$, where $\alpha(G)$ is the maximum size of an independent set of $G$. Erd\H{o}s conjectured in the 80s that for almost every graph $G$ equality holds, i.e., that for the random graph $G(n,0.5)$, $bc(G)=n-\alpha(G)$ 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 $G=G(n,0.5)$, $bc(G) \leq n-\alpha(G)-1$ with high probability. We prove a stronger bound: there exists an absolute constant $c>0$ so that $bc(G) \leq n-(1+c)\alpha(G)$ with high probability.