Extending edge colorings of distance-2 matchings in the hypercube

Casselgren, Markstörm, and Pham conjectured that any precolored distance-2 matching in the d-dimensional cube Qd 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 K₂ₘ can be extended to an edge coloring using at most $Δ(G)+1$ colors. As another generalization, we establish the same result for the Cartesian product G K1,m.