Decomposition of random graphs into complete bipartite graphs

We consider the problem of partitioning the edge set of a graph $G$ into the minimum number $\tau (G)$ of edge-disjoint complete bipartite subgraphs. We show that for a random graph $G$ in $G(n,p)$, for $p$ is a constant no greater than $1\mathrm{/}2$, almost surely $\tau (G)$ is between $n-c({\ln }_{1\mathrm{/}p}n{)}^{3+ϵ}$ and $n-2{\ln }_{1\mathrm{/}(1-p)}n$ for any positive constants $c$ and $ϵ$.