Geodesic
All vertex-transitive locally-quasiprimitive graphs have a semiregular automorphism
Journal of Algebraic Combinatorics, Tome 25 (2007) no. 2, pp. 217-232.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: The polycirculant conjecture states that every transitive 2-closed permutation group of degree at least two contains a nonidentity semiregular element, that is, a nontrivial permutation whose cycles all have the same length. This would imply that every vertex-transitive digraph with at least two vertices has a nonidentity semiregular automorphism. In this paper we make substantial progress on the polycirculant conjecture by proving that every vertex-transitive, locally-quasiprimitive graph has a nonidentity semiregular automorphism. The main ingredient of the proof is the determination of all biquasiprimitive permutation groups with no nontrivial semiregular elements.
Keywords: keywords locally-quasiprimitive graphs, semiregular automorphism, biquasiprimitive, 2-closure 
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Giudici, Michael; Xu, Jing. All vertex-transitive locally-quasiprimitive graphs have a semiregular automorphism. Journal of Algebraic Combinatorics, Tome 25 (2007) no. 2, pp. 217-232. http://geodesic.mathdoc.fr/item/JAC_2007__25_2_a1/