On a cyclic connectivity property of directed graphs

Abstract

Let us call a digraph D cycle-connected if for every pair of vertices $u,v\in V(D)$ there exists a cycle containing both u and $v$. In this paper we study the following open problem introduced by Ádám. Let D be a cycle-connected digraph. Does there exist a universal edge in D, i.e., an edge $e\in E(D)$ such that for every $w\in V(D)$ there exists a cycle C such that $w\in V(C)$ and $e\in E(C)$?

In his 2001 paper Hetyei conjectured that cycle-connectivity always implies the existence of a universal edge. In the present paper we prove the conjecture of Hetyei for bitournaments.