Factorisation of the complete graph into spanning regular factors

TL;DR

This paper studies how to break down complete graphs into regular spanning subgraphs, providing enumeration results and conjectures for general cases, which advances understanding of graph factorisations.

Contribution

It offers enumeration formulas for factorisations of complete graphs into regular factors and proposes a conjecture for the asymptotic count when the number of factors is small.

Findings

Enumeration of factorisations in specific cases

Asymptotic behaviour generalizes regular graph counts

Conjecture for general formula with few factors

Abstract

We enumerate factorisations of the complete graph into spanning regular graphs in several cases, including when the degrees of all the factors except for one or two are small. The resulting asymptotic behaviour is seen to generalise the number of regular graphs in a simple way. This leads us to conjecture a general formula when the number of factors is vanishing compared to the number of vertices.