Cyclic antiautomorphisms of Mendelsohn triple systems

Abstract

A cyclic triple ( a, b, c) is defined to be set {( a, b),( b, c),( c, a)} of ordered pairs. A Mendelsohn triple system of order v, M(2, 3, v), is a pair ( M, β), where M is a set of v points and β is a collection of cyclic triples of pairwise distinct points of M such that any ordered pair of distinct points of M is contained in precisely one cyclic triple of β. An antiautomorphism of a Mendelsohn triple system ( M, β) is a permutation of M which maps β to β -1, where β -1={( c, b, a); ( a, b, c) ϵβ}. In this paper we give necessary and sufficient conditions for the existence of a Mendelsohn triple system of order v admitting an antiautomorphism consisting of a single cycle of length d and having v - d fixed points. Further, we give a more general result for partial Mendelsohn triple systems in which the missing ordered pairs are precisely those containing two fixed points.