Covering the edges of a random graph by cliques

TL;DR

This paper investigates the minimum number of cliques needed to cover all edges in a random graph G(n,p) with constant p, establishing that this number is asymptotically proportional to n^2 divided by log^2 n.

Contribution

It provides a probabilistic analysis of the clique cover number in G(n,p), showing its asymptotic behavior for constant p, which was previously unknown.

Findings

Clique cover number is Theta(n^2 / log^2 n) with high probability.

The result characterizes the edge covering complexity in random graphs.

Provides insights into the structure of clique covers in dense random graphs.

Abstract

The clique cover number of a graph G is the minimum number of cliques required to cover the edges of graph G. In this paper we consider the random graph G(n,p), for p constant. We prove that with probability 1-o(1), the clique number of G(n,p) is Theta(n^2/\log^2n).