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

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 (D)=max16 i6 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 (D)= (D)− 1¿ 1 the better bound