Strong Fault-Hamiltonicity for the Crossed Cube and Its Extensions

Abstract

Fault-Hamiltonicity is an important measure of robustness for interconnection networks. Given a graph G = (V, E). The goal is to ensure that G − F remains Hamiltonian for every F ⊆ V ∪E such that |F|  ≤ k for some k. Obviously, one wants the best k possible. For many interconnection networks, in particular, for the crossed cube, the optimal result is known. There is a natural bound for k as the resulting graph cannot have a vertex of degree at most 1. Thus, if r is the minimum degree in G, then k ≤  r − 2. Since interconnection networks are modeled after computer networks, it is reasonable to assume that the vertex/edge faults occur following certain probability distributions. As such, it is reasonable to assume that it is unlikely for a vertex to have degree at most 1 even if r − 1 vertices/edges are deleted. Thus we may be able to delete more vertices and/or vertices if we assume that all the faults are “not clustered” at a vertex. To be precise, the goal is to find the best possible k in which G − F remains Hamiltonian for every F ⊆ V ∪E such that |F|  ≤ k and the minimum degree of G − F is at least 2. In this paper, we study this problem for the crossed cube and then extend the result to cover a large subclass of bijective connection networks as well as the more general matching composition networks.