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

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)=max1⩽i⩽n{|Vi|} the independence number of the semicomplete n-partite digraph D, where V1,V2,…,Vn 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 boundc⩾κ(D)κ(D)+1(p+1).In addition, we present examples which show that this bound is best possible.