Loopy, Hankel, and Combinatorially Skew-Hankel Tournaments

TL;DR

This paper studies structured tournaments with specific symmetries and loops, providing conditions for their existence, construction algorithms, and methods to transform between them while maintaining their properties.

Contribution

It introduces necessary and sufficient conditions, construction algorithms, and transformation switches for loopy, Hankel, and skew-Hankel tournaments.

Findings

Established existence conditions for each tournament type.

Developed algorithms for constructing these tournaments.

Designed switches for transforming tournaments within each class.

Abstract

We investigate tournaments with a specified score vector having additional structure: loopy tournaments in which loops are allowed, Hankel tournaments which are tournaments symmetric about the Hankel diagonal (the anti-diagonal), and combinatorially skew-Hankel tournaments which are skew-symmetric about the Hankel diagonal. In each case, we obtain necessary and sufficient conditions for existence, algorithms for construction, and switches which allow one to move from any tournament of its type to any other, always staying within the defined type.