Factorisation of the complete bipartite graph into spanning semiregular factors

TL;DR

This paper investigates the enumeration and asymptotic behavior of factorizations of complete bipartite graphs into spanning semiregular subgraphs, proposing conjectures and analyzing probabilities of edge-disjointness.

Contribution

It provides new enumeration results, asymptotic formulas, and conjectures for factorizations into semiregular graphs, extending understanding of their combinatorial structure.

Findings

Asymptotic enumeration of factorizations with small degrees

A conjectured general formula for the number of factorizations

Probability estimates for edge-disjoint semiregular graphs

Abstract

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