Tournaments and even graphs are equinumerous

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