Extending edge colorings of distance-3 matchings in the Cartesian product of graphs

Abstract

We investigate the problem of extending partial edge colorings in Cartesian products of graphs, with a particular focus on cases where the precolored edges form a matching. Casselgren, Granholm, and Petros conjectured that any precolored distance-3 matching in $G = C^d_{2k}$ can be extended to a $2d$-edge coloring. In this paper, we prove a theorem that implies this conjecture. Especially, our main result establishes that a precolored distance-3 matching in the Cartesian product of certain class 1 graphs can be extended to an edge coloring that uses at most as many colors as the chromatic index, provided that certain degree conditions are satisfied. In the second part of the paper, we extend these results to Cartesian products of other types of graphs as well.