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arXiv · 2605.31225

Reconfiguration graphs of $K_{2,3}$-minor-free graphs

Ruijuan Gu·Hui Lei·Zhaoxiang Li·Yulai Ma·Susu Wang
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Abstract

The $\ell$-reconfiguration graph of a graph $G$, denoted by $\mathcal{R}_{\ell}(G)$, is the graph whose vertices are the proper $\ell$-colorings of $G$, with an edge between two colorings if they differ in color on exactly one vertex. For any graph $G$ of treewidth at most $2$, Bousquet and Perarnau showed that $\mathcal{R}_\ell(G)$ has linear diameter for $\ell\geq 6$. This result was later extended by Bartier, Bousquet, and Heinrich, who proved that $\mathcal{R}_5(G)$ also has linear diameter. In this paper, we show that for each $\ell\geq 5$, the $\ell$-reconfiguration graphs of $K_{2,3}$-minor-free graphs, some of which include graphs of treewidth $3$, have linear diameter. As a key step in our proof, we also establish that the $(\ell-1)$-reconfiguration graphs of cactus graphs have linear diameter.

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Ruijuan Gu, Hui Lei, Zhaoxiang Li, Yulai Ma, Susu Wang. 2026-05-29. Reconfiguration graphs of $K_{2,3}$-minor-free graphs. https://arxiv.org/abs/2605.31225

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