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

Abstract

The Borodin-Kostochka Conjecture states that for a graph $G$, if $\Delta (G)\ge 9$, then $\chi (G)\le \max \{\Delta (G)-1,\omega (G)\}$. We use $P_t$ and $C_t$ to denote a path and a cycle on $t$ vertices, respectively. Let $C=v_1{v}_2{v}_3{v}_4{v}_5{v}_1$ be an induced $C_5$. A {\em $C_5^{+}$} is a graph obtained from $C$ by adding a $C_3=xyzx$ and a $P_2=t_1{t}_2$ such that (1) $x$ and $y$ are both exactly adjacent to $v_1,v_2,v_3$ in $V(C)$, $z$ is exactly adjacent to $v_2$ in $V(C)$, $t_1$ is exactly adjacent to $v_4,v_5$ in $V(C)$ and $t_2$ is exactly adjacent to $v_1,v_4,v_5$ in $V(C)$, (2) $t_1$ is exactly adjacent to $z$ in $\{x,y,z\}$ and $t_2$ has no neighbors in $\{x,y,z\}$. In this paper, we show that the Borodin-Kostochka Conjecture holds for ($P_6,C_4,H$)-free graphs, where $H\in \{K_7,C_5^{+}\}$. This generalizes some results of Gupta and Pradhan in \cite{GP21,GP24}.