On cycles through two arcs in strong multipartite tournaments

Abstract

A multipartite tournament is an orientation of a complete $c$-partite graph. In [L. Volkmann, A remark on cycles through an arc in strongly connected multipartite tournaments, Appl. Math. Lett. 20 (2007) 1148--1150], Volkmann proved that a strongly connected $c$-partite tournament with $c\ge 3$ contains an arc that belongs to a directed cycle of length $m$ for every $m\in \{3,4,\dots ,c\}$. He also conjectured the existence of three arcs with this property. In this note, we prove the existence of two such arcs.