Clique packings in random graphs

TL;DR

This paper investigates the maximum number of edge-disjoint near-maximal cliques in dense Erdős-Rényi random graphs, establishing matching upper and lower bounds and analyzing a sequential clique selection process.

Contribution

It proves a matching lower bound for the maximum clique packing size in dense random graphs using a novel process analysis with the Differential Equation Method.

Findings

Established a lower bound of (n^2/(log n)^3) for clique packings.

Provided a new proof of the existing upper bound for clique packings.

Discussed the open problem of the exact maximum size of such packings.

Abstract

We consider the question of how many edge-disjoint near-maximal cliques may be found in the dense Erd\H{o}s-R\'enyi random graph $G(n,p)$. Recently Acan and Kahn showed that the largest such family contains only $O(n^2/(\log{n})^3)$ cliques, with high probability, which disproved a conjecture of Alon and Spencer. We prove the corresponding lower bound, $\Omega(n^2/(\log{n})^3)$, by considering a random graph process which sequentially selects and deletes near-maximal cliques. To analyse this process we use the Differential Equation Method. We also give a new proof of the upper bound $O(n^2/(\log{n})^3)$ and discuss the problem of the precise size of the largest such clique packing.