Reconfiguration graphs for vertex colorings of $P_5$-free graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916414091165696 |
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| author | Lei, Hui Ma, Yulai Miao, Zhengke Shi, Yongtang Wang, Susu |
| author_facet | Lei, Hui Ma, Yulai Miao, Zhengke Shi, Yongtang Wang, Susu |
| contents | For any positive integer $k$, the reconfiguration graph for all $k$-colorings of a graph $G$, denoted by $\mathcal{R}_k(G)$, is the graph where vertices represent the $k$-colorings of $G$, and two $k$-colorings are joined by an edge if they differ in color on exactly one vertex. Bonamy et al. established that for any $2$-chromatic $P_5$-free graph $G$, $\mathcal{R}_k(G)$ is connected for each $k\geq 3$. On the other hand, Feghali and Merkel proved the existence of a $7p$-chromatic $P_5$-free graph $G$ for every positive integer $p$, such that $\mathcal{R}_{8p}(G)$ is disconnected.
In this paper, we offer a detailed classification of the connectivity of $\mathcal{R} _k(G) $ concerning $t$-chromatic $P_5$-free graphs $G$ for cases $t=3$, and $t\geq4$ with $t+1\leq k \leq {t\choose2}$. We demonstrate that $\mathcal{R}_k(G)$ remains connected for each $3$-chromatic $P_5$-free graph $G$ and each $k \geq 4$. Furthermore, for each $t\geq4$ and $t+1 \leq k \leq {t\choose2}$, we provide a construction of a $t$-chromatic $P_5$-free graph $G$ with $\mathcal{R}_k(G)$ being disconnected. This resolves a question posed by Feghali and Merkel. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_19368 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reconfiguration graphs for vertex colorings of $P_5$-free graphs Lei, Hui Ma, Yulai Miao, Zhengke Shi, Yongtang Wang, Susu Combinatorics 05C15 For any positive integer $k$, the reconfiguration graph for all $k$-colorings of a graph $G$, denoted by $\mathcal{R}_k(G)$, is the graph where vertices represent the $k$-colorings of $G$, and two $k$-colorings are joined by an edge if they differ in color on exactly one vertex. Bonamy et al. established that for any $2$-chromatic $P_5$-free graph $G$, $\mathcal{R}_k(G)$ is connected for each $k\geq 3$. On the other hand, Feghali and Merkel proved the existence of a $7p$-chromatic $P_5$-free graph $G$ for every positive integer $p$, such that $\mathcal{R}_{8p}(G)$ is disconnected. In this paper, we offer a detailed classification of the connectivity of $\mathcal{R} _k(G) $ concerning $t$-chromatic $P_5$-free graphs $G$ for cases $t=3$, and $t\geq4$ with $t+1\leq k \leq {t\choose2}$. We demonstrate that $\mathcal{R}_k(G)$ remains connected for each $3$-chromatic $P_5$-free graph $G$ and each $k \geq 4$. Furthermore, for each $t\geq4$ and $t+1 \leq k \leq {t\choose2}$, we provide a construction of a $t$-chromatic $P_5$-free graph $G$ with $\mathcal{R}_k(G)$ being disconnected. This resolves a question posed by Feghali and Merkel. |
| title | Reconfiguration graphs for vertex colorings of $P_5$-free graphs |
| topic | Combinatorics 05C15 |
| url | https://arxiv.org/abs/2409.19368 |