Semiregular automorphisms in vertex-transitive graphs of order \(3p^2\) (Q1753112)

Summary: 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 \textit{semiregular element}. It is the purpose of this paper to prove that vertex-transitive graphs of order \(3p^2\), where \(p\) is a prime, contain semiregular automorphisms.