Morris, Dave

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

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Browsing Morris, Dave by Author "Morris, Joy"

Now showing 1 - 5 of 5
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    Caley graphs of order 16p are hamiltonian
    (Drustvo Matematikov, Fizikov in Astronomov, 2012) Curran, Stephen J.; Morris, Dave W.; Morris, Joy
    Suppose G is a finite group, such that |G| = 16p, where p is prime. We show that if S is any generating set of G, then there is a hamiltonian cycle in the corresponding Cayley graph Cay(G;S).
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    Groups for which it is easy to detect graphical regular representations
    (University of Primorska, 2022) Morris, Dave W.; Morris, Joy; Verret, Gabriel
    We say that a finite group G is DRR-detecting if, for every subset S of G, either the Cayley digraph Cay(G,S) is a digraphical regular representation (that is, its automorphism group acts regularly on its vertex set) or there is a nontrivial group automorphism φ of G such that φ(S) = S. We show that every nilpotent DRR-detecting group is a p-group, but that the wreath product Zp wr Zp is not DRR-detecting, for every odd prime p. We also show that if G1 and G2 are nontrivial groups that admit a digraphical regular representation and either gcd(|G1|, |G2|) = 1, or G2 is not DRR-detecting, then the direct product G1 x G2 is not DRR-detecting. Some of these results also have analogues for graphical regular representations.
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    Hamiltonian cycles in Caley graphs whose order has few prime factors
    (Drustvo Matematikov, Fizikov in Astronomov, 2012) Kutnar, Klavdija; Marusic, Dragan; Morris, Dave W.; Morris, Joy; Sparl, Primoz
    We prove that if Cay(G;S) is a connected Cayley graph with n vertices,and the prime factorization of n is very small, then Cay(G;S) has a hamiltonian cycle. More precisely, if p, q, and r are distinct primes, then n can be of the form kp with 24 6= k < 32, or of the form kpq with k ≤ 5,or of the form pqr,or of the form kp2 with k ≤ 4,or of the form kp3 with k ≤ 2.
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    Non-Cayley-Isomorphic Cayley graphs from non-Cayley-Isomorphic Cayley digraphs
    (Centre for Discrete Mathematics and Computing, 2024) Morris, Dave W.; Morris, Joy
    A finite group G is said to be a non-DCI group if there exist subsets S1 and S2 of G, such that the associated Cayley digraphs C−→ay(G; S1) and C−→ay(G; S2) are isomorphic, but no automorphism of G carries S1 to S2. Furthermore, G is said to be a non-CI group if the subsets S1 and S2 can be chosen to be closed under inverses, so we have undirected Cayley graphs Cay(G; S1) and Cay(G; S2). We show that if p is a prime number, and the elementary abelian p- group (Zp)r is a non-DCI group, then (Zp)r+3 is a non-CI group. In most cases, we can also show that (Zp)r+2 is a non-CI group. In particular, from Pablo Spiga’s proof that (Z3)8 is a non-DCI group, we conclude that (Z3)10 is a non-CI group. This is the first example of a non-CI elementary abelian 3-group.
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    On colour-preserving automorphisms of Caley graphs
    (Drustvo Matematikov, Fizikov in Astronomov, 2016) Hujdurovic, Ademir; Kutnar, Klavdija; Morris, Dave W.; Morris, Joy
    We study the automorphisms of a Cayley graph that preserve its natural edge-colouring. More precisely, we are interested in groups G, such that every such automorphism of every connected Cayley graph on G has a very simple form: the composition of a left-translation and a group automorphism. We find classes of groups that have the property, and we determine the orders of all groups that do not have the property. We also have analogous results for automorphisms that permute the colours, rather than preserving them.