Extending edge-colorings of distance-2 matchings in the hypercube

Casselgren, Markstörm, and Pham conjectured that any precolored dis-tan-ce-2 matching in the $d$-dimensional cube $Q_d$ with at most $d$ colors 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 $H=G\mathrm{\square }{K}_{2m}$ with at most ${\chi }^{\mathrm{\prime }}(H)=\mathrm{\Delta }(H)=\mathrm{\Delta }(G)+2m-1$ colors can be extended to an edge-coloring using at most ${\chi }^{\mathrm{\prime }}(H)$ colors. As another generalization, we establish a similar result for the Cartesian product $G\mathrm{\square }{K}_{1,m}$.