A Critical Probability for Biclique Partition of $G_{n,p}$
Tom Bohman, Jakob Hofstad
TL;DR
This work identifies a critical constant $p_0 \approx 0.312$ for the random graph $G_{n,p}$ that marks a transition in the biclique partition number $bp(G_{n,p})$ relative to the independence number $\alpha(G_{n,p})$. Using the notions of special subgraphs, together with BKR inequalities and first/second moment methods, the authors prove that for constant $p<p_0$ whp $bp(G_{n,p}) = n - \alpha(G_{n,p})$, and for $p_0 \le p < 1/2$ whp $bp(G_{n,p}) = n - O(\alpha(G_{n,p}))$, with a sharper bound $bp(G_{n,p}) \le n - (1+c_p)\alpha(G_{n,p})$ when $p$ is strictly between $p_0$ and $1/2$. The results verify a Chung-Peng conjecture in these regimes and illuminate how the appearance of special subgraphs governs the biclique partition landscape in sparse-to-dense random graphs, while outlining key open questions near and at $p=1/2$. The approach blends careful probabilistic counting, the BKR disjoint-occurrence inequality, and the second-moment method to control the relevant subgraph structures.
Abstract
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) \leq n - α(G)$, where $α(G)$ 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 $bp(G) = n - α(G)$ 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 $bp(G_{n,p}) = n - (1 + Θ(1)) α(G_{n,p})$ with high probability.