The generalized connectivity of alternating group graphs and [formula omitted]-star graphs

Abstract

Let S⊆V(G) and κG(S) denote the maximum number r of edge-disjoint trees T1,T2,…,Tr in G such that V(Ti)⋂V(Tj)=S for any i,j∈{1,2,…,r} and i≠j. For an integer k with 2≤k≤n, the generalizedk-connectivity of a graph G is defined as κk(G)=min{κG(S)|S⊆V(G) and |S|=k}. The generalized k-connectivity is a generalization of traditional connectivity. In this paper, we focus on the alternating group graphsand (n,k)-star graphs, denoted by AGn and Sn,k, respectively. We study the generalized3-connectivity of the two kinds of graphs and show that κ3(AGn)=2n−5 for n≥4 and κ3(Sn,k)=n−2 for n≥k+1 and k≥4, which generalize the known result about star graphs given by Li et al. (2016). In addition, as the alternating group network ANn is isomorphic to Sn,k for k=n−2, the generalized 3-connectivity of ANn for n≥6 can be obtained directly.