Borodin-Kostochka conjecture for a family of $P_6$-free graphs
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916708045815808 |
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| author | Wu, Di Wu, Rong |
| author_facet | Wu, Di Wu, Rong |
| 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 $P_5$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_12062 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Borodin-Kostochka conjecture for a family of $P_6$-free graphs Wu, Di Wu, Rong Combinatorics 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 $P_5$. |
| title | Borodin-Kostochka conjecture for a family of $P_6$-free graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2306.12062 |