Title: Borodin-Kostochka conjecture for a family of $P_6$-free graphs Show Chinese title

Title: 波罗丁-科托奇卡猜想对于一类$P_6$-自由图

Abstract: Borodin and Kostochka conjectured that every graph $G$ with $\Delta\ge9$ satisfies $\chi\le$ max $\{\omega, \Delta-1\}$. Gupta and Pradhan proved the Borodin-Kostochka conjecture for ($P_5$, $C_4$)-free graphs [{\em J. Appl. Math. Comp.} \textbf{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 $P_5$. Show Chinese abstract

Abstract: 波罗丁和科托奇卡猜想，每个图$G$具有$\Delta\ge9$满足$\chi\le$最大$\{\omega, \Delta-1\}$。 古普塔和普拉德汉证明了($P_5$, $C_4$)-free 图的 Borodin-Kostochka 猜想 [{\em 应用数学与计算数学杂志} \textbf{65} (2021) 877-884]. 在本文中，我们证明了对于 ($P_6$, apple, torch )-free 图的 Borodin-Kostochka 猜想，即没有诱导的$P_6$，没有诱导的$C_5$带有一个悬挂边，以及没有诱导的$C_5$和$C_4$恰好共享一个诱导的$P_3$的图。这从允许存在$P_5$的角度推广了 Gupta 和 Pradhan 的结果。