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

For a graph $G$, let $\mathrm{\Delta }(G)$, $\omega (G)$, 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 $\mathrm{\Delta }(G)\ge 9$ and $\omega (G)<\mathrm{\Delta }(G)$ is $(\mathrm{\Delta }(G)-1)$-colorable.