Coloring $(P_6,C_4)$-free graphs with $\Delta -1$ colors

Abstract

For a graph $G$, let $\Delta (G)$, $\omega (G)$, and $\chi (G)$ denote the maximum degree, clique number, and chromatic number of $G$, respectively. Let $P_n$ and $C_n$ denote the chordless path and chordless cycle on $n$ vertices, respectively. In this paper, we prove that every $(P_6,C_4)$-free graph $G$ with $\Delta (G)\ge 9$ and $\omega (G)<\Delta (G)$ is $(\Delta (G)-1)$-colorable.