The biclique partition number of a graph $G=(V,E)$, denoted $bp(G)$, 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 $bp(G)\le n-\alpha (G)$, where $\alpha (G)$ 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 $bp(G)=n-\alpha (G)$ 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 $bp(G_{n,p})=n-(1+\mathrm{\Theta }(1))\alpha (G_{n,p})$ with high probability.

图的 biclique 分区数$G=（V,乙）$，表示为$乙p（G）$，是G的成对边不相交完全二部子图的最小数量，使得G的每条边恰好属于其中一个。很容易看出$乙p（G）\le n-\alpha （G）$， 在哪里$\alpha （G）$是独立组G的最大大小。Erdős 在 80 年代推测，对于几乎所有图G等式都成立；即，如果$G=G_{n,1/2}$然后$乙p（G）=n-\alpha （G）$有很高的概率。阿隆证明这是错误的。如果我们取而代之，我们证明 Erdős 的猜想是正确的$G=G_{n,p}$，其中p是常数且小于某个阈值$p_0\approx 0.312$。这验证了 Chung 和 Peng 对这些p值的猜想。我们还表明，如果$p_0<p<1/2$然后$乙p（G_{n,p}）=n-（1+\mathrm{\theta }（1））\alpha （G_{n,p}）$有很高的概率。