Tournaments and even graphs are equinumerous

Abstract

A graph is called odd if there is an orientation of its edges and an automorphism that reverses the sense of an odd number of its edges and even otherwise. Pontus von Brömssen (né Andersson) showed that the existence of such an automorphism is independent of the orientation and considered the question of counting pairwise non-isomorphic even graphs. Based on computational evidence, he made the rather surprising conjecture that the number of pairwise non-isomorphic even graphs on n vertices is equal to the number of pairwise non-isomorphic tournaments on n vertices. We prove this conjecture using a counting argument with several applications of the Cauchy–Frobenius theorem.