Coloring some $(P_6,C_4)$-free graphs with $Δ-1$ colors
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| Format: | Preprint |
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2024
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| _version_ | 1866913368081694720 |
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| author | Chen, Ran Wu, Di Zhang, Xiaowen |
| author_facet | Chen, Ran Wu, Di Zhang, Xiaowen |
| contents | The Borodin-Kostochka Conjecture states that for a graph $G$, if $Δ(G)\geq9$, then $χ(G)\leq\max\{Δ(G)-1,ω(G)\}$. We use $P_t$ and $C_t$ to denote a path and a cycle on $t$ vertices, respectively. Let $C=v_1v_2v_3v_4v_5v_1$ be an induced $C_5$. A {\em $C_5^+$} is a graph obtained from $C$ by adding a $C_3=xyzx$ and a $P_2=t_1t_2$ such that (1) $x$ and $y$ are both exactly adjacent to $v_1,v_2,v_3$ in $V(C)$, $z$ is exactly adjacent to $v_2$ in $V(C)$, $t_1$ is exactly adjacent to $v_4,v_5$ in $V(C)$ and $t_2$ is exactly adjacent to $v_1,v_4,v_5$ in $V(C)$, (2) $t_1$ is exactly adjacent to $z$ in $\{x,y,z\}$ and $t_2$ has no neighbors in $\{x,y,z\}$. In this paper, we show that the Borodin-Kostochka Conjecture holds for ($P_6,C_4,H$)-free graphs, where $H\in \{K_7,C_5^+\}$. This generalizes some results of Gupta and Pradhan in \cite{GP21,GP24}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_18455 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Coloring some $(P_6,C_4)$-free graphs with $Δ-1$ colors Chen, Ran Wu, Di Zhang, Xiaowen Combinatorics The Borodin-Kostochka Conjecture states that for a graph $G$, if $Δ(G)\geq9$, then $χ(G)\leq\max\{Δ(G)-1,ω(G)\}$. We use $P_t$ and $C_t$ to denote a path and a cycle on $t$ vertices, respectively. Let $C=v_1v_2v_3v_4v_5v_1$ be an induced $C_5$. A {\em $C_5^+$} is a graph obtained from $C$ by adding a $C_3=xyzx$ and a $P_2=t_1t_2$ such that (1) $x$ and $y$ are both exactly adjacent to $v_1,v_2,v_3$ in $V(C)$, $z$ is exactly adjacent to $v_2$ in $V(C)$, $t_1$ is exactly adjacent to $v_4,v_5$ in $V(C)$ and $t_2$ is exactly adjacent to $v_1,v_4,v_5$ in $V(C)$, (2) $t_1$ is exactly adjacent to $z$ in $\{x,y,z\}$ and $t_2$ has no neighbors in $\{x,y,z\}$. In this paper, we show that the Borodin-Kostochka Conjecture holds for ($P_6,C_4,H$)-free graphs, where $H\in \{K_7,C_5^+\}$. This generalizes some results of Gupta and Pradhan in \cite{GP21,GP24}. |
| title | Coloring some $(P_6,C_4)$-free graphs with $Δ-1$ colors |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2405.18455 |