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); κ G (S) refers to the maximum number of internally disjoint trees in G connecting S. The generalized k-connectivity of G, κ k (G), is defined as the minimum value of κ G (S) over all S ⊆ V (G) with |S| = k. The n-dimensional crossed cube CQ n , 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 CQ n and obtain κ 3 (CQ n ) = κ 4 (CQ n ) = n − 1, where n ≥ 2.

The generalized 4-connectivity of balanced hypercubes

“…It is NP-complete to compute κ k (G) [6]. Just a few networks' generalized 4-connectivity were determined, including hypercube [10], hierarchical cubic networks [25], exchanged hypercubes [24], divide-and-swap cube [26], pancake graphs [27], (n, k)-star networks [9], crossed cubes [11], and folded hypercubes [12]. For the n-dimensional balanced hypercube BH n , it was shown that κ 3 (BH n ) = 2n − 1 when n ≥ 1 [17].…”

The Generalized 3-Connectivity of Exchanged Folded Hypercubes

“…The generalized 3connectivities of augmented cubes, (n, k)-bubble-sort graphs, and generalized hypercubes have been obtained in [7][8][9], respectively. The generalized 4-connectivities of hypercubes, crossed cubes, exchanged hypercubes, and hierarchical cubic networks have been obtained in [10][11][12][13], respectively. On the whole, the generalized k-connectivity is known for a small number of graphs and almost all known results are about k = 3 or 4.…”