Linear recoloring diameter of degenerate chordal graphs and bounded treewidth graphs
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| Formato: | Preprint |
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2025
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| _version_ | 1866914135289102336 |
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| author | Wang, Yichen Lu, Mei |
| author_facet | Wang, Yichen Lu, Mei |
| contents | Let $G$ be a graph on $n$ vertices and $t$ an integer. The reconfiguration graph of $G$, denoted by $R_t(G)$, consists of all $t$-colorings of $G$ and two $t$-colorings are adjacent if they differ on exactly one vertex. The $t$-recoloring diameter of $G$ is the diameter of $R_t(G)$. For a $d$-degenerate graph $G$, $R_t(G)$ is connected when $t \ge d+2$~(Dyer et al., 2006). Furthermore, the $t$-recoloring diameter is $O(n^2)$ when $t \ge 3(d+1)/2$~(Bousquet et al., 2022), and it is $O(n)$ when $t \ge 2d+2$~(Bousquet and Perarnau, 2016). For a $d$-degenerate and chordal graph $G$, the $t$-recoloring diameter of $G$ is $O(n^2)$ when $t \ge d+2$~(Bonamy et al. 2014). If $G$ is a graph of treewidth at most $k$, then $G$ is also $k$-degenerate, and the previous results hold. Moreover, when $t \ge k+2$, the $t$-recoloring diameter is $O(n^2)$~(Bonamy and Bousquet, 2013). When $k=2$, the $t$-recoloring diameter of $G$ is linear when $t \ge 5$~(Bartier, Bousquet and Heinrich, 2021) and the result is tight. In this paper, we prove that if $G$ is $d$-degenerate and chordal, then the $t$-recoloring diameter of $G$ is $O(n)$ when $t \ge 2d+1$. Moreover, if the treewidth of $G$ is at most $k$, then the $t$-recoloring diameter is $O(n)$ when $t \ge 2k+1$. This result is a generalization of the previous results on graphs of treewidth at most two. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_15456 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Linear recoloring diameter of degenerate chordal graphs and bounded treewidth graphs Wang, Yichen Lu, Mei Combinatorics 05C15 Let $G$ be a graph on $n$ vertices and $t$ an integer. The reconfiguration graph of $G$, denoted by $R_t(G)$, consists of all $t$-colorings of $G$ and two $t$-colorings are adjacent if they differ on exactly one vertex. The $t$-recoloring diameter of $G$ is the diameter of $R_t(G)$. For a $d$-degenerate graph $G$, $R_t(G)$ is connected when $t \ge d+2$~(Dyer et al., 2006). Furthermore, the $t$-recoloring diameter is $O(n^2)$ when $t \ge 3(d+1)/2$~(Bousquet et al., 2022), and it is $O(n)$ when $t \ge 2d+2$~(Bousquet and Perarnau, 2016). For a $d$-degenerate and chordal graph $G$, the $t$-recoloring diameter of $G$ is $O(n^2)$ when $t \ge d+2$~(Bonamy et al. 2014). If $G$ is a graph of treewidth at most $k$, then $G$ is also $k$-degenerate, and the previous results hold. Moreover, when $t \ge k+2$, the $t$-recoloring diameter is $O(n^2)$~(Bonamy and Bousquet, 2013). When $k=2$, the $t$-recoloring diameter of $G$ is linear when $t \ge 5$~(Bartier, Bousquet and Heinrich, 2021) and the result is tight. In this paper, we prove that if $G$ is $d$-degenerate and chordal, then the $t$-recoloring diameter of $G$ is $O(n)$ when $t \ge 2d+1$. Moreover, if the treewidth of $G$ is at most $k$, then the $t$-recoloring diameter is $O(n)$ when $t \ge 2k+1$. This result is a generalization of the previous results on graphs of treewidth at most two. |
| title | Linear recoloring diameter of degenerate chordal graphs and bounded treewidth graphs |
| topic | Combinatorics 05C15 |
| url | https://arxiv.org/abs/2509.15456 |