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

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 $k<{\chi }^{\prime }(G)$ and $l<{\chi }^{\prime }(H)$, respectively, is extendable, then any edge-precoloring of $G\square H$ of size $k+l+1$ can be extended to a proper $({\chi }^{\prime }(G)+{\chi }^{\prime }(H))$-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 $H=K_2$.