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 Br\"omssen (n\'e 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 vertices is equal to the number of pairwise non-isomorphic tournaments on vertices. We prove this conjecture using a counting argument with several applications of the Cauchy-Frobenius Theorem.
Keywords
Cite
@article{arxiv.2204.01947,
title = {Tournaments and Even Graphs are Equinumerous},
author = {Gordon F. Royle and Cheryl E. Praeger and S. P. Glasby and Saul D. Freedman and Alice Devillers},
journal= {arXiv preprint arXiv:2204.01947},
year = {2023}
}
Comments
9 pages. Corrected minor non-mathematical typos and slightly expanded the introduction