Borodin-Kostochka conjecture for a family of $P_6$-free graphs

Borodin and Kostochka conjectured that every graph $G$ with $\mathrm{\Delta }\ge 9$ satisfies $\chi \le$ max $\{\omega ,\mathrm{\Delta }-1\}$. Gupta and Pradhan proved the Borodin-Kostochka conjecture for ($P_5$, $C_4$)-free graphs [{\em J. Appl. Math. Comp.} $\mathbf{\text{65}}$ (2021) 877-884]. In this paper, we prove the Borodin-Kostochka conjecture for ($P_6$, apple, torch)-free graphs, that is, graphs with no induced $P_6$, no induced $C_5$ with a hanging edge, and no induced $C_5$ and $C_4$ sharing exactly an induced $P_3$. This generalizes the result of Gupta and Pradhan from the perspective of allowing the existence of $P_5$.