Clique packings in random graphs

TL;DR

This work analyzes the size of the largest packing of near-maximal k-cliques in dense Erdős–Rényi graphs G(n,p) by embedding the problem in a random clique-removal process. The authors deploy the Differential Equation Method to track two key quantities, Q(G_m) (the number of k-cliques remaining) and Y_e(G_m) (the number of k-cliques containing a given edge), proving that these variables follow explicit trajectories with controlled error. They establish a matching lower bound of order pn^2 log n / k^4 (for k=k_0−C, C≥4) and provide a new upper bound with linear dependence on γ, improving previous exponential-type dependencies and broadening applicability. The results advance the understanding of clique packing in random graphs and illustrate the power of the differential-equation framework for unbounded clique sizes, while highlighting open questions about the exact constants and regimes.

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, $Ω(n^2/(\log{n})^3)$, 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.

Theorems & Definitions (51)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Conjecture 1.4
• Theorem 1.5
• Theorem 2.1
• proof : Proof of Theorem \ref{['thm:main']}, assuming Theorem \ref{['thm:Q']}
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4