The Generalized 3-Connectivity and 4-Connectivity of Crossed Cube

Abstract The generalized connectivity, an extension of connectivity, provides a new reference for measuring the fault tolerance of networks. For any connected graph G, let S ⊆ V (G) and 2 ≤ |S| ≤ V (G); κG(S) refers to the maximum number of internally disjoint trees in G connecting S. The generalized k-connectivity of G, κk(G), is defined as the minimum value of κG(S) over all S ⊆ V (G) with |S| = k. The n-dimensional crossed cube CQn, as a hypercube-like network, is considered as an attractive alternative to hypercube network because of its many good properties. In this paper, we study the generalized 3-connectivity and the generalized 4-connectivity of CQnand obtain κ3(CQn) = κ4(CQn) = n − 1, where n ≥ 2.