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