k-Path-Connectivity of Completely Balanced Tripartite Graphs

For a graph G = (V, E) and a set S subset of V(G) of a size at least 2, a path in G is said to be an S-path if it connects all vertices of S. Two S-paths P-1 and P-2 are said to be internally disjoint if E(P-1) boolean AND E(P-2) = circle divide and V(P-1) boolean AND V(P-2) = S; that is, they share no vertices and edges apart from S. Let pi(G)(S) denote the maximum number of internally disjoint S-paths in G. The k-path-connectivity pi(k)(G) of G is then defined as the minimum pi(G)(S), where S ranges over all k-subsets of V(G). In this paper, we study the k-path-connectivity of the complete balanced tripartite graph K-n,K-n,K-n and obtain pi(k)(K-n,K-n,K-n) = vertical bar 2n/k-1 vertical bar for 3 <= k <= n.