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Borodin and Kostochka conjectured that every graph $G$ with $\Delta\ge9$satisfies $\chi\le$ max $\{\omega, \Delta-1\}$. Gupta and Pradhan proved theBorodin-Kostochka conjecture for ($P_5$, $C_4$)-free graphs. In this note, weprove 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 two edges. This generalizes theresult of Gupta and Pradhan from the perspective of allowing the existence of$P_5$.

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