Bipartite decomposition of random graphs

TL;DR

This paper investigates the bipartite decomposition number of random graphs, disproving a long-standing conjecture for most cases and establishing bounds for different probabilities, revealing nuanced behavior of graph decompositions.

Contribution

It refutes Erdős's conjecture by showing that for most large random graphs, the bipartite decomposition number is slightly less than the conjectured value, and analyzes the behavior for sparse graphs.

Findings

For most large $n$, $ au(G) eq n-eta(G)$, but rather $ au(G) extless n-eta(G)$ by 1 or 2.

The conjecture holds only for some specific sequences of $n$, not generally.

For sparse graphs with $p extless c$, $ au(G)$ scales linearly with $eta(G)$ with high probability.

Abstract

For a graph $G=(V,E)$, let $\tau(G)$ denote 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 for every graph $G$, $\tau(G) \leq n -\alpha(G)$, where $\alpha(G)$ is the maximum size of an independent set of $G$. Erd\H{o}s conjectured in the 80s that for almost every graph $G$ equality holds, i.e., that for the random graph $G(n,0.5)$, $\tau(G)=n-\alpha(G)$ with high probability, that is, with probability that tends to $1$ as $n$ tends to infinity. Here we show that this conjecture is (slightly) false, proving that for most values of $n$ tending to infinity and for $G=G(n,0.5)$, $\tau(G) \leq n-\alpha(G)-1$ with high probability, and that for some sequences of values of $n$ tending to infinity $\tau(G) \leq n-\alpha(G)-2$ with probability bounded away from $0$. We also study the typical value of $\tau(G)$ for random graphs $G=G(n,p)$ with $p < 0.5$ and show that there is an absolute positive constant $c$ so that for all $p \leq c$ and for $G=G(n,p)$, $\tau(G)=n-\Theta(\alpha(G))$ with high probability.