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arXiv · 2209.11137

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

A. C. Burgess·P. Danziger·A. Pastine·T. Traetta
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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.

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A. C. Burgess, P. Danziger, A. Pastine, T. Traetta. 2022-09-22. Constructing uniform 2-factorizations via row-sum matrices: solutions to the Hamilton-Waterloo problem. https://arxiv.org/abs/2209.11137

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