A critical probability for biclique partition of Gn,p

Abstract

The biclique partition number of a graph $G=(V,E)$, denoted $bp\left(G\right)$, is 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 $bp\left(G\right)\le n-\alpha \left(G\right)$, where $\alpha \left(G\right)$ is the maximum size of an independent set of G. Erdős conjectured in the 80's that for almost every graph G equality holds; i.e., if $G=G_{n,1/2}$ then $bp\left(G\right)=n-\alpha \left(G\right)$ with high probability. Alon showed that this is false. We show that the conjecture of Erdős is true if we instead take $G=G_{n,p}$, where p is constant and less than a certain threshold value $p_0\approx 0.312$. This verifies a conjecture of Chung and Peng for these values of p. We also show that if $p_0<p<1/2$ then $bp\left(G_{n,p}\right)=n-(1+\Theta (1\left)\right)\alpha \left(G_{n,p}\right)$ with high probability.