CR tournaments

TL;DR

This paper introduces and studies CR tournaments, a class of tournaments with unique determinant properties, characterizing their structure and extending previous results on determinants of tournaments.

Contribution

It defines CR tournaments and strong CR tournaments, characterizes their structure, and answers an open question about determinants of specific tournament classes.

Findings

All $L_n$ are strong CR tournaments.

CR tournaments are characterized by switching equivalence to transitive blowups.

The paper answers an open question from prior research.

Abstract

The determinant of a tournament $T$ is defined as the determinant of the skew-adjacency matrix of $T$. For a positive odd integer $k$, let $\mathcal{D}_k$ be the set of tournaments whose all subtournaments have determinant at most $k^2$. Some existing results show that, for $k \in \{1,3,5\}$, a tournament $T \in \mathcal{D}_k \backslash \mathcal{D}_{k-2}$ ($T \in \mathcal{D}_1$ when $k=1$) if and only if $T$ is switching equivalent to a transitive blowup of $L_{k+1}$, where $L_{k+1}$ is a tournament of order $k+1$ with a specific structure. There exist some tournaments with the special property that adding any vertex that does not conform to their structure increases the maximum value of determinants among their subtournaments. We define these tournaments as CR tournaments. In this paper, we introduce CR tournaments, strong CR tournaments and basic tournaments, and show some properties and conclusions on these tournaments. For a basic strong CR tournament $H \in \mathcal{D}_{k} \backslash \mathcal{D}_{k-2}$, we show that if $T$ contains a subtournament which is switching isomorphic to $H$, then $T \in \mathcal{D}_{k} \backslash \mathcal{D}_{k-2}$ if and only if $T$ is switching equivalent to a transitive blowup of $H$. Moreover, we demonstrate that all $L_n$ are strong CR tournaments, and based on this conclusion, we answer a question posed in [J. Zeng, L. You, On determinants of tournaments and $\mathcal{D}_k$, arXiv:2408.06992, 2024.], and propose some questions for further research.