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

Abstract

For a graph $G$, let $Δ(G)$, $ω(G)$, and $χ(G)$ 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 $Δ(G)\ge 9$ and $ω(G)<Δ(G)$ is $(Δ(G)-1)$-colorable.