The 3-path-connectivity of the star graphs

Abstract

Let G be a graph with vertex set V(G) and S⊆V(G). An S-path of G is a path which connects all vertices of S in G. Let P1 and P2 be S-paths. If E(P1)∩E(P2)=0̸ and V(P1)∩V(P2)=S, then the two paths P1 and P2 are called internally disjointS-paths. Denoted by πG(S) the maximum number of any pairwise internally disjoint S-paths in G. The r-path-connectivityπr(G) is defined as πr(G)= min{πG(S)|S⊆V(G),|S|=r}, where 2≤r≤|V(G)|. In this paper, we focus on the 3-path-connectivity of the n-dimensional star graph Sn and obtain that π3(Sn)=3(n−1)−14 with n≥3.