Every cycle-connected multipartite tournament has a universal arc

A digraph D is cycle-connected if for every pair of vertices u , v ∈ V ( D ) there exists a directed cycle in D containing both u and v . In 1999, Ádám [A. Ádám, On some cyclic connectivity properties of directed graphs, Acta Cybernet. 14 (1) (1999) 1–12] posed the following problem. Let D be a cycle-connected digraph. Does there exist a universal arc in D , i.e., an arc e ∈ A ( D ) such that for every vertex w ∈ V ( D ) there is a directed cycle in D containing both e and w ? artite or multipartite tournament is an orientation of a complete c -partite graph. Recently, Hubenko [A. Hubenko, On a cyclic connectivity property of directed graphs, Discrete Math. 308 (2008) 1018–1024] proved that each cycle-connected bipartite tournament has a universal arc. As an extension of this result, we show in this note that each cycle-connected multipartite tournament has a universal arc.