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

TL;DR

This paper investigates Walecki tournaments with a unique arc in a directed triangle, establishing fixed points under automorphisms and analyzing the prevalence of such structures among all signatures.

Contribution

It proves the existence of a fixed vertex under automorphisms for certain Walecki tournaments and analyzes the distribution of signatures producing such tournaments.

Findings

Existence of a fixed vertex in specific Walecki tournaments.

Automorphisms map the special vertex to itself.

At least half of the signatures produce tournaments with the unique triangle property.

Abstract

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).