CR tournaments

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 $\mathcal{D}<em>k$ be the set of tournaments whose all subtournaments have determinant at most $k<sup>2$. Some existing results show that, for $k \in {1,3,5}$, a tournament $$T \in \mathcal{D}_k \backslash \mathcal{D}</em>{k-2}$$ ($T \in \mathcal{D}<em>1$ when $k=1$) if and only if $T$ is switching equivalent to a transitive blowup of $L</em>{k+1}$, 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 $$H \in \mathcal{D}<em>{k} \backslash \mathcal{D}</em>{k-2}$$, we show that if $T$ contains a subtournament which is switching isomorphic to $H$, then $$T \in \mathcal{D}<em>{k} \backslash \mathcal{D}</em>{k-2}$$ 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.