Vu's conjecture holds for claw-free graphs
Linda Cook, Ross J. Kang, Eileen Robinson, Gabriëlle Zwaneveld
TL;DR
The paper addresses Vu's conjecture, linking the chromatic number χ(G) to the maximum codegree Δ2(G) in claw-free graphs, and proves χ(G) ≤ Δ2(G)+3 for this class with tightness shown via the line graph of the Petersen graph. The authors adopt a structural, defuzzification-based strategy rooted in the King–Reed framework, first resolving the quasi-line graph case and then extending to all claw-free graphs by ruling out G_cf- and G_ql-critical counterexamples across a comprehensive set of graph families and 2-join configurations. Key technical contributions include handling homogeneous pairs of cliques, thickenings, and canonical interval 2-joins, plus detailed treatment of antiprismatic, three-cliqued, and Icosahedral thickenings. They also extend the result to the edge-codegree setting via Δ_e and discuss implications for list coloring and broader Vu-type conjectures, outlining future directions and potential extensions.
Abstract
Given a graph $G$, let $Δ_2(G)$ denote the maximum number of neighbors any two distinct vertices of $G$ have in common. Vu (2002) proposed that, provided $Δ_2(G)$ is not too small as a proportion of the maximum degree $Δ(G)$ of $G$, the chromatic number of $G$ should never be too much larger than $Δ_2(G)$. We make a first approach towards Vu's conjecture from a structural graph theoretic point of view. We prove that, in the case where $G$ is claw-free, indeed the chromatic number of $G$ is at most $Δ_2(G)+3$. This is tight, as our bound is met with equality for the line graph of the Petersen graph. Moreover, we can prove this in terms of the more specific parameter that bounds the maximum number of neighbors any two endpoints of some edge of $G$ have in common. Our result may be viewed as a generalization of the classic bound of Vizing (1964) for edge-coloring.