Constructing uniform 2-factorizations via row-sum matrices: solutions to the Hamilton-Waterloo problem

TL;DR

This paper introduces row-sum matrices over non-abelian groups and uses them to solve many open cases of the Hamilton-Waterloo problem by constructing uniform 2-factorizations of graphs.

Contribution

It extends the concept of row-sum matrices to generalized dihedral groups and applies them to construct new uniform 2-factorizations solving numerous open cases.

Findings

Constructed row-sum matrices over non-abelian groups.

Solved many open cases of the Hamilton-Waterloo problem.

Expanded the spectrum of known graph factorizations.

Abstract

In this paper, we formally introduce the concept of a row-sum matrix over an arbitrary group $G$. When $G$ is cyclic, these types of matrices have been widely used to build uniform 2-factorizations of small Cayley graphs (or, Cayley subgraphs of blown-up cycles), which themselves factorize complete (equipartite) graphs. Here, we construct row-sum matrices over a class of non-abelian groups, the generalized dihedral groups, and we use them to construct uniform $2$-factorizations that solve infinitely many open cases of the Hamilton-Waterloo problem, thus filling up large parts of the gaps in the spectrum of orders for which such factorizations are known to exist.