Extending partial edge-colorings of bounded size in Cartesian products of graphs
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| Format: | Preprint |
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2026
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| _version_ | 1866911566643855360 |
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| author | Bärnkopf, Pál Győri, Ervin |
| author_facet | Bärnkopf, Pál Győri, Ervin |
| contents | 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 $k<χ'(G)$ and $l<χ'(H)$, respectively, is extendable, then any edge-precoloring of $G \square H$ of size $k+l+1$ can be extended to a proper $(χ'(G)+χ'(H))$-coloring. We provide partial progress toward this conjecture by establishing the result in cases where $k<Δ(G)$, $G$ 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 $H = K_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23139 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Extending partial edge-colorings of bounded size in Cartesian products of graphs Bärnkopf, Pál Győri, Ervin Combinatorics 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 $k<χ'(G)$ and $l<χ'(H)$, respectively, is extendable, then any edge-precoloring of $G \square H$ of size $k+l+1$ can be extended to a proper $(χ'(G)+χ'(H))$-coloring. We provide partial progress toward this conjecture by establishing the result in cases where $k<Δ(G)$, $G$ 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 $H = K_2$. |
| title | Extending partial edge-colorings of bounded size in Cartesian products of graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.23139 |