The generalized 3-connectivity of two kinds of regular networks

The locally twisted cube LTQn and the generalized hypercube GQ(m1, m2, ⋯,mn) are two important kinds of regular graphs in the modelling interconnection networks. For measuring the fault tolerance and reliability of interconnection networks, the generalized k-connectivity was introduced, which is a generalization of the traditional connectivity. More precisely, let S⊆V(G) and κG(S) denote the maximum number r of trees T1,T2,⋯,Tr such that V(Ti)∩V(Tj)=S and E(Ti)∩E(Tj)=∅ for any integers 1≤i≠j≤r. For an integer k with 2≤k≤|V(G)|, the generalized k-connectivity of G, denoted by κk(G), is defined as κk(G)=min⁡{κG(S)|S⊆V(G)and|S|=k}. In this paper, we study the generalized 3-connectivity of the locally twisted cube LTQn and the generalized hypercube GQ(m1,m2,⋯,mn), and get two main results that κ3(LTQn)=n−1 and κ3(GQ(m1,m2,⋯,mn))=∑i=1n(mi−1)−1. © 2021 Elsevier B.V.