List-coloring claw-free graphs with $\Delta$-1 colors

TL;DR

This paper proves that certain claw-free graphs with high maximum degree can be list-colored with one fewer colors than their maximum degree, advancing understanding of coloring properties in these graphs.

Contribution

It establishes a new bound for list-coloring claw-free graphs with high maximum degree, extending previous results to a broader class of graphs.

Findings

List-coloring of quasi-line graphs with $igtriangleup(G)>igomega(G)$ and $igtriangleup(G) extgreater 69$ is achievable with $igtriangleup(G)-1$ colors.

For claw-free graphs with $igtriangleup(G)>igomega(G)$ and $igtriangleup(G) extgreater 69$, the list-chromatic number is at most $igtriangleup(G)-1$.

The result improves bounds on coloring claw-free graphs with high maximum degree.

Abstract

We prove that if $G$ is a quasi-line graph with $\Delta(G)>\omega(G)$ and $\Delta(G)\ge 69$, then $\chi_{OL}(G)\le \Delta(G)-1$. Together with our previous work, this implies that if $G$ is a claw-free graph with $\Delta(G)>\omega(G)$ and $\Delta(G)\ge 69$, then $\chi_{\ell}(G)\le \Delta(G)-1$.