Cayley numbers with arbitrarily many distinct prime factors

A positive integer n is a Cayley number if every vertex-transitive graph of order n is a Cayley graph. In 1983, Dragan MaruÅ¡ič posed the problem of determining the Cayley numbers. In this paper we give an infinite set S of primes such that every finite product of distinct elements from S is a Cayley number. This answers a 1996 outstanding question of Brendan McKay and Cheryl Praeger, which they “believe to be the key unresolved question” on Cayley numbers. We also show that, for every finite product n of distinct elements from S, every transitive group of degree n contains a semiregular element.