Enumerating graphs via even/odd dichotomy

Abstract

Following Hujdurović et al. (2016), an automorphism of a graph is said to be even/odd if it acts on the vertex set of the graph as an even/odd permutation. In this paper the formula for calculating the number of graphs of order n admitting odd automorphisms and the formula for calculating the number of graphs of order n without odd automorphisms are given together with their asymptotic estimates.Such numbers are also considered for the subclass of vertex-transitive graphs. A positive integer n is a Cayley number if every vertex-transitive graph of order n is a Cayley graph. In analogy, a positive integer n is said to be a vertex-transitive-odd number (in short, a VTO-number) if every vertex-transitive graph of order n admits an odd automorphism. It is proved that there exists infinitely many VTO numbers which are square-free and have arbitrarily long prime factorizations. Further, it is proved that Cayley numbers congruent to 2 modulo 4, cubefree nilpotent Cayley numbers congruent to 3 modulo 4, and numbers of the form 2p, p a prime, are VTO numbers. At the other extreme, it is proved that for a positive integer n the complete graph Kn and its complement are the only vertex-transitive graphs of order n admitting odd automorphisms if and only if n is a Fermat prime.