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Permutation groups, vertex-transitive graphs and semiregular automorphisms
Marušič, Dragan ; Scapellato, Raffaele
A nonidentity element of a permutation group id said to be semiregular if all of its orbits have the same lenght. The work in this paper is linked to some earlier papers where the problem of ...existence of semiregular automorphisms in vertex-transitive digraphs was posed. It vas observed there that every vertex-transitive digraph of order ▫$p^k$▫ or ▫$mp$▫, where ▫$p$▫ is a prime, ▫$k \ge 1$▫ ans ▫$m \le p$▫ are posutive integers, has a semiregular automorphism. On the other hand, there are transitive permutation groups without semiregular elements. In this paper, it is proved that every cubic vertex- transitive graph contains a semiregular automorphism, and moreover, it is shown that every vertex-transitive diagraph of order ▫$2p^2$▫, where ▫$p$▫ is a prime, contains a semiregular automorphism.
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