A note on the cyclic connectivity property of directed graphs - Chunling, Tong; Xiaohui, Lin; Yuansheng, Yang; Feng, Xue; Huapingc, Jiang (Utilitas Mathematica, 2011)
A digraph D is cycle-connected if for every pair of vertices u, v ε; V(D) there exists a directed cycle in D containing both u and v.A.Hubenko [On a cyclic connectivity property of directed graphs, Discrete Math.308 (2008) 1018-1024] proved that each cycle-connected bitournament has a universal arc, and further raised the following problem.Assume that D is a cycle-connected bitournament and C is a maximal cycle of D.Are all arcs of C universal? In the present paper, we show that there exists a cycle-connected bipartite tournament such that at least one arc of one of its maximal cycles is not universal.More over, we show that there exists a simple bipartite cycle-connected digraph such that at least one arc of one of its maximal cycles is not universal, and that there exists a simple bipartite cycle-connected digraph such that all arcs of its maximal cycles are universal.