Some Additional Cases of Bicyclic Antiautomorphisms of Mendelsohn Triple Systems

Abstract

A cyclic triple, $(a,b,c)$, is defined to be the set $\{(a,b),(b,c),(c,a)\}$ of ordered pairs. A Mendelsohn triple system of order $v$, or MTS$(v)$, is a pair $(M,\beta )$, where $M$ is a set of $v$ points and $\beta$ is a collection of cyclic triples, each containing pairwise distinct points of $M$ such that every ordered pair of distinct points of $M$ exists in exactly one cyclic triple of $\beta$. An antiautomorphism of a Mendelsohn triple system $(M,\beta )$ is a permutation of $M$ which maps $\beta$ to ${\beta }^{-1}$, where ${\beta }^{-1}=\{(c,b,a)\mid (a,b,c)\in \beta \}$. Necessary conditions for the existence of an MTS$(v)$ admitting an antiautomorphism consisting of two cycles of lengths $M$ and $N$, where $1<M\le N$, have been shown, and for the cases of $N=M$ and $N=2M$, sufficiency has been shown. We show sufficiency for the cases in which $M=13$ and $N=78,390,$ and $702$.