Semiregular automorphisms in vertex-transitive graphs with a solvable group of automorphisms - Marušič, Dragan (Ars mathematica contemporanea, 2017)
It has been conjectured that automorphism groups of vertex-transitive (di)graphs, and more generally 2-closures of transitive permutation groups, must necessarily possess a fixed-point-free element of prime order, and thus a non-identity element with all orbits of the same length, in other words, a semiregular element. The known affirmative answers for graphs with primitive and quasiprimitive groups of automorphisms suggest that solvable groups need to be considered if one is to hope for a complete solution of this conjecture. It is the purpose of this paper to present an overview of known results and suggest possible further lines of research towards a complete solution of the problem.
Keywords: avtomorfizem brez negibnih točk; policirkulantna domneva; polregularni avtomorfizem; rešljiva grupa