A Critical Probability for Biclique Partition of $G_{n,p}$

TL;DR

This paper investigates the biclique partition number of Erdős–Rényi random graphs, proving a phase transition at a specific probability threshold where the conjectured equality holds, and characterizing the behavior above this threshold.

Contribution

It establishes the validity of Erdős's conjecture for random graphs with constant probability below a threshold and describes the biclique partition number's asymptotic behavior above it.

Findings

Erdős's conjecture holds for G_{n,p} when p < p_0 ≈ 0.312.

For p > p_0, the biclique partition number is approximately n minus a scaled independence number.

The paper confirms a conjecture of Chung and Peng for certain probability ranges.

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 - \alpha(G)$, where $\alpha(G)$ is the maximum size of an independent set of $G$. Erd\H{o}s 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 Erd\H{o}s 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 $bp(G_{n,p}) = n - (1 + \Theta(1)) \alpha(G_{n,p})$ with high probability.