Edge Clique Cover of Claw-free Graphs

TL;DR

This paper proves that all connected claw-free graphs with independence number at least three have an edge clique cover number at most equal to their number of vertices, characterizing the cases of equality.

Contribution

It extends the known bound on edge clique cover number from line and quasi-line graphs to all claw-free graphs with independence number at least three, using the structure theorem.

Findings

For connected claw-free graphs with independence number ≥ 3, cc(G) ≤ n.

Equality holds only for specific graphs: icosahedron, twister, or p-th power of cycle.

Provides a complete characterization of extremal graphs for the bound.

Abstract

The smallest number of cliques, covering all edges of a graph $ G $, is called the (edge) clique cover number of $ G $ and is denoted by $ cc(G) $. It is an easy observation that for every line graph $ G $ with $ n $ vertices, $cc(G)\leq n $. G. Chen et al. [Discrete Math. 219 (2000), no. 1--3, 17--26; MR1761707] extended this observation to all quasi-line graphs and questioned if the same assertion holds for all claw-free graphs. In this paper, using the celebrated structure theorem of claw-free graphs due to Chudnovsky and Seymour, we give an affirmative answer to this question for all claw-free graphs with independence number at least three. In particular, we prove that if $ G $ is a connected claw-free graph on $ n $ vertices with $ \alpha(G)\geq 3 $, then $ cc(G)\leq n $ and equality holds if and only if $ G $ is either the graph of icosahedron, or the complement of a graph on $10$ vertices called twister or the $p^{th}$ power of the cycle $ C_n $, for $1\leq p \leq \lfloor (n-1)/3\rfloor $.