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arXiv · 2609.28893

Yes, $(2K_2, K_4)$-free graphs are recolorable

Henry Echeverría·Owen Henderschedt
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Abstract

We prove that every $(2K_2,K_4)$-free graph is recolorable. Equivalently, for every such graph $G$ and every $\ell\geq χ(G)+1$, the reconfiguration graph of proper $\ell$-colorings of $G$, in which two colorings are adjacent if they differ on exactly one vertex, is connected. This resolves the final remaining open case in the classification of recolorable $(F_1,F_2)$-free graphs when $F_1$ and $F_2$ have at most four vertices.

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Henry Echeverría, Owen Henderschedt. 2026-09-24. Yes, $(2K_2, K_4)$-free graphs are recolorable. https://arxiv.org/abs/2609.28893

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