Morris, Joy

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Browsing Morris, Joy by Author "Morris, Joy"

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    Automorphism groups of wreath product digraphs
    (Electronic Journal of Combinatorics, 2009) Dobson, Edward; Morris, Joy
    We generalize a classical result of Sabidussi that was improved by Hemminger, to the case of directed color graphs. The original results give a necessary and sufficient condition on two graphs, C and D, for the automorphsim group of the wreath product of the graphs, Aut(C o D) to be the wreath product of the automorphism groups Aut(C) o Aut(D). Their characterization generalizes directly to the case of color graphs, but we show that there are additional exceptional cases in which either C or D is an infinite directed graph. Also, we determine what Aut(C o D) is if Aut(C o D) 6= Aut(C) o Aut(D), and in particular, show that in this case there exist vertex-transitive graphs C0 and D0 such that C0 oD0 = C oD and Aut(C oD) = Aut(C0) o Aut(D0).
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    Automorphisms of circulants that respect partitions
    (University of Calgary, Department of Mathematics & Statistics, 2016) Morris, Joy
    In this paper, we begin by partitioning the edge (or arc) set of a circulant (di)graph according to which generator in the connection set leads to each edge. We then further refine the partition by subdividing any part that corresponds to an element of order less than n, according to which of the cycles generated by that element the edge is in. It is known that if the (di)graph is connected and has no multiple edges, then any automorphism that respects the first partition and fixes the vertex corresponding to the group identity must be an automorphism of the group (this is in fact true in the more general context of Cayley graphs). We show that automorphisms that respect the second partition and fix 0 must also respect the first partition, and so are again precisely the group automorphisms of Zn.
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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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    Cayley graphs of more than one abelian group
    (University of Primorska, 2021) Dobson, Edward; Morris, Joy
    We show that for certain integers n, the problem of whether or not a Cayley digraph Γ of ℤn is also isomorphic to a Cayley digraph of some other abelian group G of order n reduces to the question of whether or not a natural subgroup of the full automorphism group contains more than one regular abelian group up to isomorphism (as opposed to the full automorphism group). A necessary and sufficient condition is then given for such circulants to be isomorphic to Cayley digraphs of more than one abelian group, and an easy-to-check necessary condition is provided.
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    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.
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    Characterising CCA Sylow cyclic groups whose order is not divisible by four
    (Drustvo Matematikov, Fizikov in Astronomov, 2018) Morgan, Luke; Morris, Joy; Verret, Gabriel
    A Cayley graph on a group G has a natural edge-colouring. We say that such a graph is CCA if every automorphism of the graph that preserves this edge-colouring is an element of the normaliser of the regular representation of G. A group G is then said to be CCA if every connected Cayley graph on G is CCA. Our main result is a characterisation of non-CCA graphs on groups that are Sylow cyclic and whose order is not divisible by four. We also provide several new constructions of non-CCA graphs.
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    The CI problem for infinite groups
    (Electronic Journal of Combinatorics, 2016) Morris, Joy
    A finite group G is a DCI-group if, whenever S and S0 are subsets of G with the Cayley graphs Cay(G,S) and Cay(G,S0) isomorphic, there exists an automorphism ϕ of G with ϕ(S) = S0. It is a CI-group if this condition holds under the restricted assumption that S = S−1. We extend these definitions to infinite groups, and make two closely-related definitions: an infinite group is a strongly (D)CIf-group if the same condition holds under the restricted assumption that S is finite; and an infinite group is a (D)CIf-group if the same condition holds whenever S is both finite and generates G. We prove that an infinite (D)CI-group must be a torsion group that is not locallyfinite. We find infinite families of groups that are (D)CIf-groups but not strongly (D)CIf-groups, and that are strongly (D)CIf-groups but not (D)CI-groups. We discuss which of these properties are inherited by subgroups. Finally, we completely characterise the locally-finite DCI-graphs on Zn. We suggest several open problems related to these ideas, including the question of whether or not any infinite (D)CIgroup exists.
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    Classification of vertex-transitive digraphs of order a product of two distinct primes via automorphism group
    (2025) Dobson, Ted; Hujdurovic, Ademir; Kutnar, Klavdija; Morris, Joy
    In the mid-1990s, two groups of authors independently obtained classifications of vertex-transitive graphs whose order is a product of two distinct primes. In the intervening years it has become clear that there is additional information concerning these graphs that would be useful, as well as making explicit the extensions of these results to digraphs. Additionally, there are several small errors in some of the papers that were involved in this classification. The purpose of this paper is to fill in the missing information as well as correct all known errors.
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    Cyclic m-cycle systems of complete graphs minus a 1-factor
    (The University of Queensland, Centre for Discrete Mathematics and Computing, 2017) Jordan, Heather; Morris, Joy
    In this paper, we provide necessary and sufficient conditions for the existence of a cyclic m-cycle system of Kn −I when m and n are even and m | n.
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    Detecting graphical and digraphical regular representations in groups of squarefree order
    (2025) Morris, Joy; Verret, Gabriel
    A necessary condition for a Cayley digraph Cay(R,S) to be a regular representation is that there are no non-trivial group automorphisms of R that fix S setwise. A group is DRR-detecting or GRR-detecting if this condition is also sufficient for all Cayley digraphs or graphs on the group, respectively. In this paper, we determine precisely which groups of squarefree order are DRR detecting, and which are GRR-detecting.
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    Digraphs with small automorphism groups that are Cayley on two nonisomorphic groups
    (University of Primorska, 2020) Morgan, Luke; Morris, Joy; Verret, Gabriel
    Let Γ = Cay(G, S) be a Cayley digraph on a group G and let A = Aut(Γ). The Cayley index of Γ is |A : G|. It has previously been shown that, if p is a prime, G is a cyclic p-group and A contains a noncyclic regular subgroup, then the Cayley index of Γ is superexponential in p. We present evidence suggesting that cyclic groups are exceptional in this respect. Specifically, we establish the contrasting result that, if p is an odd prime and G is abelian but not cyclic, and has order a power of p at least p3, then there is a Cayley digraph Γ on G whose Cayley index is just p, and whose automorphism group contains a nonabelian regular subgroup.
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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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    Groups with elements of order 8 do not have the DCI property
    (University of Primorska, 2025) Dobson, Ted; Morris, Joy; Spiga, Pablo
    Let k be odd, and n an odd multiple of 3. Although this can also be deduced from known results, we provide a new proof that Ck ⋊ C₈ and (Cn × C₃) ⋊ C₈ do not have the Directed Cayley Isomorphism (DCI) property. When k is prime, Ck ⋊ C₈ had previously been proved to have the Cayley Isomorphism (CI) property. To the best of our knowledge, the groups Cp ⋊ C₈ (where p is an odd prime) are only the second known infinite family of groups that have the CI property but do not have the DCI property. This also provides a new proof of the result (which follows from known results but was not explicitly published) that no group with an element of order 8 has the DCI property. One piece of our proof is a new result that may prove to be of independent interest: we show that if a permutation group has a regular subgroup of index 2 then it must be 2-closed.
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    Haar graphical representations of finite groups and an application to poset representations
    (Elsevier, 2025) Morris, Joy; Spiga, Pablo
    Answering a question of Feng, Kovács, Wang, and Yang, we classify the finite groups admitting a Haar graphical representation. Specifically, we show that every finite group admits a Haar graphical representation, with abelian groups and ten other small groups as the only exceptions. Our work on Haar graphs allows us to improve a 1980 result of Babai concerning representations of groups on posets, achieving the best possible result in this direction. An improvement to Babai's related result on representations of groups on distributive lattices follows.
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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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    Most generalized Petersen graphs of girth 8 have cop number 4
    (Centre for Discrete Mathematics and Computing, 2022) Morris, Joy; Runte, Tigana; Skelton, Adrian
    A generalized Petersen graph GP (n, k) is a regular cubic graph on 2n vertices (the parameter k is used to define some of the edges). It was previously shown (Ball et al., 2015) that the cop number of GP (n, k) is at most 4, for all permissible values of n and k. In this paper we prove that the cop number of “most” generalized Petersen graphs is exactly 4. More precisely, we show that unless n and k fall into certain specified categories, then the cop number of GP (n, k) is 4. The graphs to which our result applies all have girth 8. In fact, our argument is slightly more general: we show that in any cubic graph of girth at least 8, unless there exist two cycles of length 8 whose intersection is a path of length 2, then the cop number of the graph is at least 4. Even more generally, in a graph of girth at least 9 and minimum valency δ, the cop number is at least δ +1
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    Most rigid representations and Cayley index
    (University of Primorska, 2018) Morris, Joy; Tymburski, Josh
    For any finite group G, a natural question to ask is the order of the smallest possible automorphism group for a Cayley graph on G. A particular Cayley graph whose automorphism group has this order is referred to as an MRR (Most Rigid Representation), and its Cayley index is a numerical indicator of this value. Study of GRRs showed that with the exception of two infinite families and thirteen individual groups, every group admits a Cayley graph whose MRR is a GRR, so that the Cayley index is 1. The full answer to the question of finding the smallest possible Cayley index for a Cayley graph on a fixed group was almost completed in previous work, but the precise answers for some finite groups and one infinite family of groups were left open. We fill in the remaining gaps to completely answer this question.
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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.
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    On generalised Petersen graphs of girth 7 that have cop number 4
    (University of Primorska, 2022) Morris, Harmony; Morris, Joy
    We show that if n = 7k/i with i ∈ {1, 2, 3} then the cop number of the generalised Petersen graph GP(n,k) is 4, with some small previously-known exceptions. It was previously proved by Ball et al. (2015) that the cop number of any generalised Petersen graph is at most 4. The results in this paper explain all of the known generalised Petersen graphs that actually have cop number 4 but were not previously explained by Morris et al. in a recent preprint, and places them in the context of infinite families. (More precisely, the preprint by Morris et al. explains all known generalised Petersen graphs with cop number 4 and girth 8, while this paper explains those that have girth 7.)