Cycles through arcs in multipartite tournaments and a conjecture of Volkmann

Volkmann [L. Volkmann, A remark on cycles through an arc in strongly connected multipartite tournaments, Appl. Math. Lett. 20 (2007) 1148–1150] conjectured that a strong c -partite tournament with c ≥ 3 contains three arcs that belong to a cycle of length m for each m ∈ { 3 , 4 , … , c } . In this work, we prove that Volkmann’s conjecture is true.