Extending partial edge colorings of iterated cartesian products of cycles and paths

We consider the problem of extending partial edge colorings of iterated cartesian products of even cycles and paths, focusing on the case when the precolored edges satisfy either an Evans-type condition or is a matching. In particular, we prove that if G=Cᵈ₂₊ is the dth power of the cartesian product of the even cycle C₂₊ with itself, and at most 2d-1 edges of G are precolored, then there is a proper 2d-edge coloring of G that agrees with the partial coloring. We show that the same conclusion holds, without restrictions on the number of precolored edges, if any two precolored edges are at distance at least 4 from each other. For odd cycles of length at least 5, we prove that if G=Cᵈ₂₊+₁ is the dth power of the cartesian product of the odd cycle C₂₊+₁ with itself (k2), and at most 2d edges of G are precolored, then there is a proper (2d+1) -edge coloring of G that agrees with the partial coloring. Our results generalize previous ones on precoloring extension of hypercubes Journal of Graph Theory 95 (2020) 410--444.