On determinants of tournaments and $\mathcal{D}_k$

TL;DR

This paper characterizes the set of tournaments with all subtournaments having determinants bounded by a specific odd square, extending previous results and providing constructions for tournaments with exact determinant values.

Contribution

It characterizes the set _5, explores properties of _k, and constructs tournaments with determinant k^2 for any odd k, showing these sets are non-empty.

Findings

Characterization of _5.

Properties of _k for odd k.

Existence of tournaments with determinant k^2 for all odd k.

Abstract

Let $T$ be a tournament with $n$ vertices $v_1,\ldots,v_n$. The skew-adjacency matrix of $T$ is the $n\times n$ zero-diagonal matrix $S_T = [s_{ij}]$ in which $s_{ij}=-s_{ji}=1$ if $ v_i $ dominates $ v_j $. We define the determinant $\det(T)$ of $ T $ as the determinant of $ S_T $. It is well-known that $\det(T)=0$ if $n$ is odd and $\det(T)$ is the square of an odd integer if $n$ is even. Let $\mathcal{D}_k$ be the set of tournaments whose all subtournaments have determinant at most $ k^{2} $, where $k$ is a positive odd integer. The necessary and sufficient condition for $T\in \mathcal{D}_1$ or $T\in \mathcal{D}_3$ has been characterized in $2023$. In this paper, we characterize the set $\mathcal{D}_5$, obtain some properties of $\mathcal{D}_k$. Moreover, for any positive odd integer $k$, we give a construction of a tournament $T$ satisfying that $\det(T)=k^2$, and $T\in \mathcal{D}_k\backslash\mathcal{D}_{k-2}$ if $k\geq 3$, which implies $\mathcal{D}_k\backslash\mathcal{D}_{k-2}$ is not an empty set for $k\geq 3$.