Lethbridge Number Theory and Combinatorics Seminar: Ted Dobson

In 1983, Maru\v si\v c posed the problem of determining which positive integers $n$ have the property that every vertex-transitive graph of order $n$ is isomorphic to a Cayley graph of some group.  Such an integer $n$ is called a Cayley number.  Much work on this problem has been done, and, for example, it is known exactly which integers divisible by a square are Cayley numbers.  These are $p^2$, $p^3$, and $12$.  Additionally, a fair amount is known via constructions about which square-free integers are not Cayley numbers.  Much less is known about which square-free integers are Cayley numbers, and it is not even known if there is a Cayley number that is a product of five distinct primes.  We answer a question posed by C.~Praeger who asked if there was a Cayley number of order $n$ where $n$ has $k$ distinct prime factors for every positive integer $k$.  We construct an infinite set of distinct prime numbers $S$ with the property that the product of any $k$ elements of $S$ is always a Cayley number.  This is joint work with Pablo Spiga.