Walecki tournaments with an arc that lies in a unique directed triangle

A Walecki tournament is any tournament that can be formed by choosing an orientation for each of the Hamilton cycles in the Walecki decomposition of a complete graph on an odd number of vertices. In this paper, we show that if some arc in a Walecki tournament on at least $7$ vertices lies in exactly one directed triangle, then there is a vertex of the tournament (the vertex typically labelled $*$ in the decomposition) that is fixed under every automorphism of the tournament. Furthermore, any isomorphism between such Walecki tournaments maps the vertex labelled $*$ in one to the vertex labelled $*$ in the other. We also show that among Walecki tournaments with a signature of even length $2k$, of the $2^{2k}$ possible signatures, at least $2^k$ produce tournaments that have an arc that lies in a unique directed triangle (and therefore to which our result applies).