Morris, Dave

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https://opus.uleth.ca/handle/10133/5246

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Now showing 1 - 5 of 11
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    Cayley graphs on nilpotent groups with cyclic commutator subgroup are hamiltonian
    (University of Primorska, 2014) Ghaderpour, Ebrahim; Morris, Dave W.
    We show that if G is any nilpotent, finite group, and the commutator subgroup of G is cyclic, then every connected Cayley graph on G has a hamiltonian cycle.
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    Odd-order Cayley graphs with commutator subgroup of order pq are hamiltonian
    (University of Primorska, 2015) Morris, Dave W.
    We show that if G is a nontrivial, finite group of odd order, whose commutator subgroup [G, G] is cyclic of order pμqν, where p and q are prime, then every connected Cayley graph on G has a hamiltonian cycle.
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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.
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    Cayley graphs on groups with commutator subgroup of order 2p are hamiltonian
    (University of Primorska, 2018) Morris, Dave W.
    We show that if G is a finite group whose commutator subgroup [G, G] has order 2p, where p is an odd prime, then every connected Cayley graph on G has a hamiltonian cycle.
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    Automorphisms of the canonical double cover of a toroidal grid
    (University of Primorska, 2023) Morris, Dave W.
    The Cartesian product of two cycles (of length m and length n) has a natural embedding on the torus, such that each face of the embedding is a 4-cycle. The toroidal grid Qd(m,n,r) is a generalization of this in which there is a shift by r when traversing the meridian of length m. In 2008, Steve Wilson found two interesting infinite families of (nonbipartite) toroidal grids that are unstable. (By definition, this means that the canonical bipartite double cover of the grid has more than twice as many automorphisms as the grid has.) It is easy to see that bipartite grids are also unstable, because the canonical double cover is disconnected. Furthermore, there are degenerate cases in which there exist two different vertices that have the same neighbours. This paper proves Wilson's conjecture that Qd(m,n,r) is stable for all other values of the parameters. In addition, we prove an analogous conjecture of Wilson for the triangular grids Tr(m,n,r) that are obtained by adding a diagonal to each face of Qd(m,n,r) (with all of the added diagonals parallel to each other).