Decomposition of random graphs into complete bipartite graphs

TL;DR

This paper investigates how to decompose random graphs into the fewest complete bipartite subgraphs, providing bounds on the minimum number needed for graphs in the Erdős–Rényi model with constant probability.

Contribution

It establishes probabilistic bounds on the minimum number of complete bipartite subgraphs needed to cover the edges of a random graph in G(n,p).

Findings

Almost sure bounds for $ au(G)$ in G(n,p)

Bounds depend on parameters n, p, c, and epsilon

Results hold for constant p ≤ 1/2

Abstract

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