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Browsing Faculty Research and Publications by Author "Dobson, Ted"

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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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    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.