Squared chromatic number without claws or large cliques
Wouter Cames van Batenburg, Ross J. Kang

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 be a claw-free graph on vertices with clique number , and consider the chromatic number of the square of . Writing for the supremum of over the line graphs of simple graphs of maximum degree at most , we prove that for . For , this implies the sharp bound . For , this implies , which is within 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.
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