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.
Citation
Royle , G F , Praeger , C E , Glasby , S P , Freedman , S D & Devillers , A 2023 , ' Tournaments and even graphs are equinumerous ' , Journal of Algebraic Combinatorics , vol. 57 , pp. 515-524 . https://doi.org/10.1007/s10801-022-01197-0
Publication
Journal of Algebraic Combinatorics
Status
Peer reviewed
ISSN
0925-9899Type
Journal article
Description
Funding: SDF was supported by a St Leonard’s International Doctoral Fees Scholarship and a School of Mathematics & Statistics PhD Funding Scholarship at the University of St Andrews. SPG was supported by the Australian Research Council Discovery Project DP190100450.Collections
Items in the St Andrews Research Repository are protected by copyright, with all rights reserved, unless otherwise indicated.
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