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
Rights
Copyright © The Author(s) 2022. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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.
Related items
Showing items related by title, author, creator and subject.
-
Automorphism groups of countable algebraically closed graphs and endomorphisms of the random graph
Dolinka, Igor; Gray, Robert Duncan; McPhee, Jillian Dawn; Mitchell, James David; Quick, Martyn (2016-05) - Journal articleWe establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph R. As a consequence we show that, for any countable graph Γ, there are uncountably many maximal subgroups ... -
Criterion of unrecognizability of a finite group by its Gruenberg–Kegel graph
Cameron, Peter J.; Maslova, Natalia (2022-10-01) - Journal articleThe Gruenberg--Kegel graph Γ(G) associated with a finite group G is an undirected graph without loops and multiple edges whose vertices are the prime divisors of |G| and in which vertices p and q are adjacent in Γ(G) if ... -
Endomorphisms of Fraïssé limits and automorphism groups of algebraically closed relational structures
McPhee, Jillian Dawn (University of St Andrews, 2012-11-30) - ThesisLet Ω be the Fraïssé limit of a class of relational structures. We seek to answer the following semigroup theoretic question about Ω. What are the group H-classes, i.e. the maximal subgroups, of End(Ω)? Fraïssé limits for ...