On Maximum Edge Cuts of Connected Digraphs

A set F of edges in a digraph D is called a directed cut if there exists a nontrivial partition (X,Y) of V(D) such that F consists of all directed edges from X to Y. Let Λ(D) denote the maximum size of a directed cut of D, and let D(1,1) be the set of all digraphs D such that d+(v)=1 or d-(v)=1 for any vertex v in D. We show that Λ(D)>= 38(|E(D)|-1) for any connected digraph D is an element of D(1,1), which provides a positive answer to a problem of Lehel, Maffray, and Preissmann. Additionally, we consider triangle-free digraphs in D(1,1) and answer another question of theirs.