📝 SummarXiv

Abstract

Casselgren, Markst\"orm, and Pham conjectured that any precolored distance-2 matching in the $d$-dimensional cube $Q_d$ can be extended to a proper $d$-edge coloring. In this paper, we prove this conjecture and some related theorems. Especially, our result establishes that if $G$ is a bipartite graph, then a precolored distance-2 matching in the Cartesian product $G \square K_{2m}$ can be extended to an edge coloring using at most $\Delta(G)+1$ colors. As another generalization, we establish the same result for the Cartesian product $G \square K_{1,m}$.