A Fault-Free Unicast Algorithm in the Generalized Hypercube with Restricted Faulty Vertices

The generalized hypercube is a highly-connective network proposed by Bhuyan and Agrawal. In this paper, we study the restricted connectivity and fault-free unicast algorithm in the generalized hypercube. Firstly, we prove that the restricted connectivity of a [Formula: see text]-dimensional generalized hypercube [Formula: see text] is [Formula: see text], where [Formula: see text] denotes the connectivity of [Formula: see text] and [Formula: see text] max [Formula: see text], under the condition that each vertex has at least one fault-free neighbor. It is almost twice the traditional connectivity of [Formula: see text]. Furthermore, we present an [Formula: see text] fault-free unicast algorithm in the generalized hypercube.

Lineark-Arboricity in Product Networks

A linear k-forest is a forest whose components are paths of length at most k. The linear k-arboricity of a graph G, denoted bylak(G), is the least number of linear k-forests needed to decompose G. Recently, Zuo, He, and Xue studied the exact values of the linear(n−1)-arboricity of Cartesian products of various combinations of complete graphs, cycles, complete multipartite graphs. In this paper, for general k we show thatmax{lak(G),lal(H)}≤lamax{k,l}(G□H)≤lak(G)+lal(H)for any two graphs G and H. Denote byG∘H, G×HandG⊠Hthe lexicographic product, direct product and strong product of two graphs G and H, respectively. For any two graphs G and H, we also derive upper and lower bounds oflak(G∘H),lak(G×H)andlak(G⊠H)in this paper. The linear k-arboricity of a 2-dimensional grid graph, a r-dimensional mesh, a r-dimensional torus, a r-dimensional generalized hypercube and a hyper Petersen network are also studied.