2024/07/04 by Joy Morris, Morris, Joy
Computer Science · #05C20 #05C38 #05C45 #05C60 #Artificial Intelligence in Games #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2407.03807
openalex publication_date 2024/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 22k possible signatures, at least 2k produce tournaments that have an arc that lies in a unique directed triangle (and therefore to which our result applies).