Two characterizations of simple circulant tournaments

TL;DR

This paper characterizes simple circulant tournaments by exploring their acyclic disconnection properties, proving they are keen with a maximum disconnection of two, and establishing equivalences related to their structure.

Contribution

It generalizes previous results on circulant tournaments, proves they are keen with specific disconnection values, and links structural properties to aperiodicity.

Findings

Circulant tournaments are $ ightarrow$keen with disconnection value 2.

The disconnection measures $ ightarrow$equal for acyclic and triangle-free cases.

Structural properties like simplicity and aperiodicity are equivalent in this context.

Abstract

The \textit{acyclic disconnection} $\overrightarrow{\omega }(D)$ (resp. the \textit{directed triangle free disconnection } $\overrightarrow{\omega }_{3}(D)$) of a digraph $D$ is defined as the maximum possible number of connected components of the underlying graph of $D\setminus A(D^{\ast })$ where $D^{\ast }$ is an acyclic (resp. a directed triangle free) subdigraph of $D$. In this paper, we generalize some previous results and solve some problems posed by V. Neumann-Lara (The acyclic disconnection of a digraph, Discrete Math. 197/198 (1999), 617-632). Let $\overrightarrow{C}_{2n+1}(J)$ be a circulant tournament. We prove that $\overrightarrow{C}_{2n+1}(J)$ is $\overrightarrow{% \omega }$-keen and $\overrightarrow{\omega _{3}}$-keen, respectively, and $% \overrightarrow{\omega }(\overrightarrow{C}_{2n+1}(J))=\overrightarrow{% \omega }_{3}(\overrightarrow{C}_{2n+1}(J))=2$ for every $\overrightarrow{C}% _{2n+1}(J)$. Finally, it is showed that $\overrightarrow{\omega }_{3}(% \overrightarrow{C}_{2n+1}(J))=2$, $\overrightarrow{C}_{2n+1}(J)$ is simple and $J$ is aperiodic are equivalent propositions.