Automorphisms and Enumeration of Switching Classes of Tournaments

The main result of this paper is a characterization of the abstract finite groups which are full automorphism groups of switching classes of tournaments: they are those whose Sylow 2-subgroups are cyclic or dihedral. Unlike previous results of this type, we do not give an explicit construction, but only an existence proof. However, the proof gives additional information; for example, if G has cyclic or dihedral Sylow 2-subgroups, then there is a switching class C, with Aut(C) = G, such that every subgroup of G of odd order is the full automorphism group of some tournament in C. We also find conditions for a permutation group to be contained in the automorphism group of some switching class. Applying this result to individual permutations leads to an enumeration of switching classes, of switching classes admitting odd permutations, and of tournaments in a switching class. 1 Introduction The concept of switching of graphs (sometimes referred to as Seidel equivalence) was first defined..