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Vertex-transitive digraphs with extra automorphisms that preserve the natural arc-colouring [Elektronski vir]
Dobson, Edward Tauscher, 1965- ...
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 auto- morphisms 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.
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