Two characterizations of simple circulant tournaments

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 $\overrightarrow{% ω}$-keen and $\overrightarrow{{\omega }_3}$-keen, respectively, and for every $\overrightarrow{C}$. Finally, it is showed that ${\overrightarrow{\omega }}_3($, ${\overrightarrow{C}}_{2n+1}(J)$ is simple and $J$ is aperiodic are equivalent propositions.