Maximum Directed Cuts in Graphs with Degree Constraints

The Max Cut problem is an NP-hard problem and has been studied extensively. Alon et al. (J Graph Theory 55:1–13, 2007) studied a directed version of the Max Cut problem and observed its connection to the Hall ratio of graphs. They proved, among others, that if an acyclic digraph has m edges and each vertex has indegree or outdegree at most 1, then it has a directed cut of size at least 2m/5. Lehel et al. (J Graph Theory 61:140–156, 2009) extended this result by replacing the “acyclic digraphs” with the “digraphs containing no directed triangles”. In this paper, we characterize the acyclic digraphs with m edges whose maximum dicuts have exactly 2m/5 edges, and our approach gives an alternative proof of the result of Lehel et al. We also show that there are infinitely many positive rational numbers β < 2/5 for which there exist digraphs D (with directed triangles) such that each vertex of D has indegree or outdegree at most 1, and any maximum directed cut in D has size precisely β|E(D)|.