The generalized 3-connectivity.of equally complete k-partite graph and its line graph

Abstract

For a vertex set S with cardinality at least 2 in a graph G, we need a tree in order to connected the set, where this tree is usually called a Steiner tree connecting S (or an S  tree). Two Steiner trees T and T  are said to be internally disjoint if V(T) V (T)  S and E(T)
E(T)  . Let ( ) G  S denote the maximum number of internally disjoint Steiner trees connecting S inG. The generalized k-connectivity ( ) k  G of a graphG which was introduced by Chartand et al. (1984) and defined as: ( ) min{ ( ) : ( ) and }. k G  G   S S V G S  k In this paper. we determine the .generalized .3-connectivity of equally complete k -partite graph and its line graphs.