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

Abstract: The Borodin-Kostochka Conjecture states that for a graph $G$, if $\Delta(G)\geq9$, then $\chi(G)\leq\max{\Delta(G)-1,\omega(G)}$. We use $P_t$ and $C_t$ to denote a path and a cycle on $t$ vertices, respectively. Let $C=v_1v_2v_3v_4v_5v_1$ be an induced $C_5$. A {\em $C_5^+$} is a graph obtained from $C$ by adding a $C_3=xyzx$ and a $P_2=t_1t_2$ 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}.