The ratio of the longest cycle and longest path in semicomplete multipartite digraphs

Abstract

A digraph obtained by replacing each edge of a complete n-partite graph by an arc or a pair of mutually opposite arcs is called a semicomplete n-partite digraph. We call α( D)=max 1⩽ i⩽ n {| V i |} the independence number of the semicomplete n-partite digraph D, where V 1, V 2,…, V n are the partite sets of D. Let p and c, respectively, denote the number of vertices in a longest directed path and the number of vertices in a longest directed cycle of a digraph D. Recently, Gutin and Yeo proved that c⩾( p+1)/2 for every strongly connected semicomplete n-partite digraph D. In this paper we present for the special class of semicomplete n-partite digraphs D with connectivity κ( D)= α( D)−1⩾1 the better bound c⩾ κ(D) κ(D)+1 (p+1). In addition, we present examples which show that this bound is best possible.