Factorisation of the Complete Bipartite Graph into Spanning Semiregular Factors

We enumerate factorisations of the complete bipartite graph into spanning semiregular graphs in several cases, including when degrees all factors except one or two are small. The resulting asymptotic behavior is seen to generalize number an elegant way. This leads us conjecture a general formula vanishing compared vertices. As corollary, we find average ways partition edges random subgraphs cases. Our proof case uses switching argument probability that set sufficiently sparse edge-disjoint randomly labeled.