Clique covers of H-free graphs

TL;DR

This paper investigates the minimum number of cliques needed to cover edges in graphs excluding certain bipartite subgraphs, providing bounds and addressing open questions in graph theory.

Contribution

It establishes upper bounds on clique covers for H-free graphs and shows that even graphs with no large stable set may require many cliques, resolving two open problems.

Findings

O(|G|^{2-1/(2t)}) clique cover bound for H-free graphs

Existence of graphs with no large stable set requiring many cliques

Addresses open questions from recent conference discussions

Abstract

It takes $n^2/4$ cliques to cover all the edges of a complete bipartite graph $K_{n/2,n/2}$, but how many cliques does it take to cover all the edges of a graph $G$ if $G$ has no $K_{t,t}$ induced subgraph? We prove that $O(|G|^{2-1/(2t)})$ cliques suffice; and also prove that, even for graphs with no stable set of size four, we may need more than linearly many cliques. This settles two questions discussed at a recent conference in Lyon.