Squared chromatic number without claws or large cliques

TL;DR

This paper establishes new bounds on the chromatic number of the square of claw-free graphs with small clique numbers, advancing understanding of graph coloring in these special classes and approaching conjectured limits.

Contribution

It proves upper bounds for the chromatic number of squared claw-free graphs with clique numbers 3 and 4, relating them to line graph parameters and near the conjectured optimal bounds.

Findings

For $ ext{clique number}=3$, $ ext{chromatic number} ext{ of } G^2 ext{ is at most } 10.

For $ ext{clique number}=4$, $ ext{chromatic number} ext{ of } G^2 ext{ is at most } 22.

Bounds are close to the conjectured optimal bounds, advancing the theory of graph coloring.

Abstract

Let $G$ be a claw-free graph on $n$ vertices with clique number $\omega$, and consider the chromatic number $\chi(G^2)$ of the square $G^2$ of $G$. Writing $\chi'_s(d)$ for the supremum of $\chi(L^2)$ over the line graphs $L$ of simple graphs of maximum degree at most $d$, we prove that $\chi(G^2)\le \chi'_s(\omega)$ for $\omega \in \{3,4\}$. For $\omega=3$, this implies the sharp bound $\chi(G^2) \leq 10$. For $\omega=4$, this implies $\chi(G^2)\leq 22$, which is within $2$ of the conjectured best bound. This work is motivated by a strengthened form of a conjecture of Erd\H{o}s and Ne\v{s}et\v{r}il.