Math & Computer Science

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Arc-disjoint hamiltonian paths in Cartesian products of directed cycles
(University of Primorska, 2025) Darijani, Iren; Miraftab, Babak; Morris, Dave W.
We show that if C1 and C2 are directed cycles (of length at least two), then the Cartesian product C1 □ C2 has two arc-disjoint hamiltonian paths. (This answers a question asked by J. A. Gallian in 1985.) The same conclusion also holds for the Cartesian product of any four or more directed cycles (of length at least two), but some cases remain open for the Cartesian product of three directed cycles. We also discuss the existence of arc-disjoint hamiltonian paths in 2-generated Cayley digraphs on (finite or infinite) abelian groups.
Cayley graphs on abelian and generalized dihedral groups
(University of Primorska, 2023) Morris, Joy; Skelton, Adrian
A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group. In this paper we give conditions for when a Cayley graph on an abelian group can be represented as a Cayley graph on a generalized dihedral group, and conditions for when the converse is true.