Cyclic uniform 2-factorizations of the complete multipartite graph

TL;DR

This paper studies cyclic uniform 2-factorizations of complete multipartite graphs, providing a complete solution for cases where the 2-factors are unions of cycles of length 4 or the entire graph, advancing the understanding of the generalized Oberwolfach Problem.

Contribution

It offers a complete characterization of cyclic uniform 2-factorizations for the extremal cases in the generalized Oberwolfach Problem, specifically for cycles of length 4 and the entire graph.

Findings

Complete solutions for 2-factorizations with cycles of length 4.

Complete solutions for 2-factorizations with cycles covering the entire graph.

Advances in the understanding of the generalized Oberwolfach Problem.

Abstract

The generalization of the Oberwolfach Problem, proposed by J. Liu in 2000, asks for a uniform $2$-factorization of the complete multipartite graph $K_{m\times n}$. Here we focus our attention on $2$-factorizations regular under the cyclic group $Z_{mn}$, whose $2$-factors are disjoint union of cycles all of even length $\ell$. In particular, we present a complete solution for the extremal cases $\ell=4$ and $\ell=mn$.