Every Cycle-Connected Multipartite Tournament with δ ≥ 2 Contains At Least Two Universal Arcs

A digraph D = (V(D), A(D)) is called cycle-connected if for every pair of vertices $${u, v\in V(D)}$$ there exists a cycle containing both u and v. Adam (Acta Cybernet 14(1):1---12, 1999) proposed the question: Let D be a cycle-connected digraph. Does there exist a universal arc in D, i.e., an arc $${e\in A(D)}$$ such that for every vertex $${w\in V(D)}$$ there exists a cycle C in D containing both e and w?. Recently, Lutz Volkmann and Stefan Winzen have proved that this conjecture is true for multipartite tournaments. As an improvement of this result, we show in this note that every cycle-connected multipartite tournament with ? ? 2 contains at least two universal arcs.