# Faculty Research and Publications

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### Browsing Faculty Research and Publications by Subject "Caley"

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- ItemCharacterising CCA Sylow cyclic groups whose order is not divisible by four(Drustvo Matematikov, Fizikov in Astronomov, 2018) Morgan, Luke; Morris, Joy; Verret, GabrielA 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.
- ItemThe CI problem for infinite groups(Electronic Journal of Combinatorics, 2016) Morris, JoyA 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.
- ItemToida's conjecture is true(Electronic Journal of Combinatorics, 2002) Dobson, Edward; Morris, JoyLet 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.
- ItemVertex-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, JoyIn 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.