Cycles through Large Degree Vertices in Digraphs: A Generalization of Meyniel's Theorem

Abstract

Let D =( V , E ) be a digraph with vertex set V of size n and arc set E . For u ∈ V , let d ( u ) denote the degree of u . A Meyniel set M is a subset of V such that d ( u )+ d ( v )⩾2 n −1 for every pair of nonadjacent vertices u and v belonging to M . In this paper we show that if D is strongly connected, then every Meyniel set M lies in a (directed) cycle, generalizing Meyniel's theorem. Our proof yields an O (| M | n 4 ) algorithm for finding such a cycle. As a corollary it follows that if D is strongly connected, then D contains a cycle through all vertices of degree ⩾ n which generalizes a result of Shi for undirected graphs.