Cayley numbers with arbitrarily many distinct prime factors

TL;DR

This paper identifies an infinite set of primes such that any product of distinct primes from this set yields a Cayley number, advancing understanding of the structure of Cayley numbers and their relation to vertex-transitive graphs.

Contribution

The paper constructs an infinite set of primes where all finite products are Cayley numbers, resolving a longstanding open problem in the classification of Cayley numbers.

Findings

Every finite product of distinct primes from the set is a Cayley number.

Every transitive group of degree equal to such a product contains a semiregular element.

Answers an outstanding question about the structure of Cayley numbers.

Abstract

A positive integer $n$ is a Cayley number if every vertex-transitive graph of order $n$ is a Cayley graph. In 1983, Dragan Maru\v{s}i\v{c} 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.