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In this paper, we formally introduce the concept of a row-sum matrix over anarbitrary group $G$. When $G$ is cyclic, these types of matrices have beenwidely 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, thegeneralized dihedral groups, and we use them to construct uniform$2$-factorizations that solve infinitely many open cases of theHamilton-Waterloo problem, thus filling up large parts of the gaps in thespectrum of orders for which such factorizations are known to exist.

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