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arXiv · 2603.23139

Extending partial edge-colorings of bounded size in Cartesian products of graphs

Pál Bärnkopf·Ervin Győri
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Abstract

This paper studies edge-precoloring extensions in Cartesian products of graphs, motivated by a conjecture of Casselgren, Petros, and Fufa. We formulate a general hypothesis stating that if every edge-precoloring of $G$ and $H$ of sizes $k<\chi'(G)$ and $l<\chi'(H)$, respectively, is extendable, then any edge-precoloring of $G \square H$ of size $k+l+1$ can be extended to a proper $(\chi'(G)+\chi'(H))$-coloring. We provide partial progress toward this conjecture by establishing the result in cases where $k<\Delta(G)$, $G$ is a triangle-free $r$-regular graph and $H$ is a star, an even cycle, a path or, more generally, an arbitrary tree $F$. Furthermore, we prove the conjecture in the case where $G$ is a subcubic graph and $H = K_2$.

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Pál Bärnkopf, Ervin Győri. 2026-03-24. Extending partial edge-colorings of bounded size in Cartesian products of graphs. https://arxiv.org/abs/2603.23139

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