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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.12062 |
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Table of Contents:
- Borodin and Kostochka conjectured that every graph $G$ with $Δ\ge9$ satisfies $χ\le$ max $\{ω, Δ-1\}$. 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 $P_5$.