vertex-transitive graphs admitting semiregular automorphisms

In 1981 maruˇsiˇc asked whether there exists a vertex-transitive digraph without a non-identity automorphism having all of its orbits of the same length. by powering a group element up, the existence ofsuch an automorphism (which is usually called semiregular) is equivalent to the existence of a fixed-pointfree automorphism of prime order. in 1988, independently, the above problem was again proposed byjordan. in the 15th british combinatorial conference, in 1995, klin proposed a more general question inthe context of 2-closed groups; is there a 2-closed transitive permutation group containing no fixed-free element of prime order? broadly speaking, klin’s question is a graph colored version of the maruˇsiˇc-jordanquestion. a graph admitting a fixed-point free automorphism of prime order is called polycirculant, so it iscustomary to refer to the conjecture that every 2-closed transitive permutation group admits a fixed-pointfree element of prime order as the polycirculant conjecture. while there has been a lot of works on thisconjecture and some of its variants, it is still wide open. in this lecture we talk about recent trends andopen problems relating to the conjecture and review some applications of semiregular automorphisms.