Clique packings in random graphs

Abstract

We consider the question of how many edge-disjoint near-maximal cliques may be found in the dense Erdős-Rényi 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, $\hat{I}\text{c◯}(n^2/(\log n{)}^3)$, 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.

45 pages