Finite cubic graphs admitting a cyclic group of automorphism with at most three orbits on vertices |
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Authors: | Primož Potočnik Micael Toledo |
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Abstract: | The theory of voltage graphs has become a standard tool in the study of graphs admitting a semiregular group of automorphisms. We introduce the notion of a cyclic generalised voltage graph to extend the scope of this theory to graphs admitting a cyclic group of automorphisms that may not be semiregular. We use this new tool to classify all cubic graphs admitting a cyclic group of automorphisms with at most three vertex-orbits and we characterise vertex-transitivity for each of these classes. In particular, we show that a cubic vertex-transitive graph admitting a cyclic group of automorphisms with at most three orbits on vertices either belongs to one of 5 infinite families or is isomorphic to the well-known Tutte–Coxeter graph. |
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Keywords: | Voltage graph Cubic graph Vertex-transitive Cyclic group Few orbits |
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