Morris, Joy

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https://opus.uleth.ca/handle/10133/4941

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Browsing Morris, Joy by Subject "Caley"

Now showing 1 - 5 of 5
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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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    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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    Toida's conjecture is true
    (Electronic Journal of Combinatorics, 2002) Dobson, Edward; Morris, Joy
    Let S be a subset of the units in Zn. Let Γ be a circulant graph of order n (a Cayley graph of Zn) such that if ij ∈ E(Γ), then i − j (mod n) ∈ S. Toida conjectured that if Γ0 is another circulant graph of order n, then Γ and Γ 0 are isomorphic if and only if they are isomorphic by a group automorphism of Zn. In this paper, we prove that Toida’s conjecture is true. We further prove that Toida’s conjecture implies Zibin’s conjecture, a generalization of Toida’s conjecture.
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    Vertex-transitive digraphs with extra automorphisms that preserve the natural arc-colouring
    (The University of Queensland, Centre for Discrete Mathematics and Computing, 2017) Dobson, Ted; Hujdurovic, Ademir; Kutnar, Klavdija; Morris, Joy
    In a Cayley digraph on a group G, if a distinct colour is assigned to each arc-orbit under the left-regular action of G, it is not hard to show that the elements of the left-regular action of G are the only digraph automorphisms that preserve this colouring. In this paper, we show that the equivalent statement is not true in the most straightforward generalisation to G-vertex-transitive digraphs, even if we restrict the situation to avoid some obvious potential problems. Specifically, we display an infinite family of 2-closed groups G, and a G-arc-transitive digraph on each (without any digons) for which there exists an automorphism of the digraph that is not an element of G (it is an automorphism of G). Since the digraph is G-arc-transitive, the arcs would all be assigned the same colour under the colouring by arc-orbits, so this digraph automorphism is colour-preserving.