Hamiltonian Cycle and Path Embeddings in 3-Ary 3-Cubes Based on K1,3 Structure Faults

Abstract

The k-ary n-cube is one of the most attractive interconnection networks for parallel and distributed computing systems. In this paper, we investigate Hamiltonian cycle and path embeddings in 3-ary 3-cubes based on K1,3-structure faults, which means each faulty element is isomorphic to any connected subgraph of a connected graph K1,3. We show that for two arbitrary distinct healthy nodes of a faulty 3-ary 3-cube, there exists a fault-free Hamiltonian path connecting these two nodes if there exists at most one faulty element that is isomorphic to any connected subgraph of K1,3. We also show that there exists a fault-free Hamiltonian cycle if the number of faulty elements are at most 2 and each faulty element is isomorphic to any connected subgraph of K1, 3.