Permutations and cubic graphs
Joel L. Brenner, Roger C. Lyndon · Pacific Journal of Mathematics · 1983
In studying maximal nonparabolic subgroups of the modular group, B. H. Neumann and later C. Tretkoff were led to study pairs of permutations A and B of an infinite set Ω such that A 2 = B 3 = 1 and that C = AB is transitive on Ω.Here we study such triples (Ω, A, B), but without the requirement that Ω be infinite.Our method is to associate with each such triple a graph G*(Ω, A, B).Such graphs have been used before, especially by Stothers and by Cori.Our central result here is that the graphs under consideration are precisely those that can be obtained by attaching trees, in certain simple specified ways, to finite or infinite graphs equipped with a reduced path that traverses each edge exactly once in each direction.