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Quasi-semiregular automorphisms of cubic and tetravalent arc-transitive graphs


  • Istvan KOVACS

Primary authors


A non-trivial automorphism $g$ of a graph $\Gamma$ is called semiregular if the only power $g^i$ fixing a vertex is the identity mapping, and it is called quasi-semiregular if it fixes one vertex and the only power $g^i$ fixing another vertex is the identity mapping. In this paper, we prove that $K_4,$ the Petersen graph and the Coxeter graph are the only connected cubic arc-transitive graphs admitting a quasi-semiregular automorphism, and $K_5$ is the only connected tetravalent $2$-arc-transitive graph admitting a quasi-semiregular automorphism. It will also be shown that every connected tetravalent $G$-arc-transitive graph, where $G$ is a solvable group containing a quasi-semiregular automorphism, is a normal Cayley graph of an abelian group of odd order.