Decomposition of random graphs into complete bipartite graphs

Abstract

We consider the problem of partitioning the edge set of a graph $G$ into the minimum number $τ(G)$ 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 $1/2$, almost surely $τ(G)$ is between $n- c(\ln_{1/p} n)^{3+ε}$ and $n - 2\ln_{1/(1-p)} n$ for any positive constants $c$ and $ε$.