Two characterizations of simple circulant tournaments

The \textit{acyclic disconnection} $\overrightarrow{ω}(D)$ (resp. the \textit{directed triangle free disconnection } $\overrightarrow{ω}_{3}(D)$) 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 $\overrightarrow{ω_{3}}$-keen, respectively, and $$ for every $\overrightarrow{C}$. Finally, it is showed that $\overrightarrow{ω}_{3}($, $\overrightarrow{C}_{2n+1}(J)$ is simple and $J$ is aperiodic are equivalent propositions.

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