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Semiregular automorphisms of arc-transitive graphs with valency \(pq\). (English) Zbl 1142.05044

A semiregular automorphism of a graph is a nonidentity automorphism all orbits of which are of equal length. Marušič has conjectured that every vertex-transitive digraph has a semiregular automorphism [D. Marušič, Discrete Math. 36, 69–81 (1981; Zbl 0459.05041)]. This conjecture remains open despite several partial results. In this paper it is shown that if a graph is arc-transitive, has valency \(pq\) (\(p,q\) not necessarily distinct primes) and its automorphism group has a nonabelian minimal normal subgroup with at least three orbits on the vertex set, then it does have a semiregular automorphism. The proof uses properties of elusive permutation groups which are transitive permutation groups containing no fixed point free elements of prime order.

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)

Citations:

Zbl 0459.05041
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References:

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