Bipartite decomposition of random graphs

For a graph $G=(V,E)$, let $\tau(G)$ 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 $G$, $\tau(G) \leq n -\alpha(G)$, 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 $G(n,0.5)$, $\tau(G)=n-\alpha(G)$ 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 most values of $n$ tending to infinity and for $G=G(n,0.5)$, $\tau(G) \leq n-\alpha(G)-1$ with high probability, and that for some sequences of values of $n$ tending to infinity $\tau(G) \leq n-\alpha(G)-2$ with probability bounded away from $0$. We also study the typical value of $\tau(G)$ 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 $p \leq c$ and for $G=G(n,p)$, $\tau(G)=n-\Theta(\alpha(G))$ with high probability.