Orthogonal matchings

AbstractConsider a graph G which is a union of m spanning subgraphs regular of degree k. We show that for k ⩾ 3 there is a matching of size m which uses exactly one edge from each subgraph. A problem of Alspach asks whether this is true for k = 2. We find a matching of size m − m23 (for large m) when k = 2, using at most one edge from each subgraph and for k = 1 we get a matching of size m − 32m23 (for large m). For subgraphs regular of degree 1 (i.e. perfect matchings) and G being the complete bipartite graph Km,m a matching with one edge from each factor corresponds to a transversal in a Latin square