Cycles through Large Degree Vertices in Digraphs: A Generalization of Meynielʹs Theorem

LetD=(V, E) be a digraph with vertex setVof sizenand arc setE. Foru∈V, letd(u) denote the degree ofu. AMeyniel set Mis a subset ofVsuch thatd(u)+d(v)⩾2n−1 for every pair of nonadjacent verticesuandvbelonging toM. In this paper we show that ifDis strongly connected, then every Meyniel setMlies in a (directed) cycle, generalizing Meynielʹs theorem. Our proof yields anO(|M| n4) algorithm for finding such a cycle. As a corollary it follows that ifDis strongly connected, thenDcontains a cycle through all vertices of degree ⩾nwhich generalizes a result of Shi for undirected graphs.