Fault Hamiltonicity of Meshes with Two Wraparound Edges

We consider the fault hamiltonian properties of m×n meshes with two wraparound edges in the first row and the last row, denoted by M2(m, n), m ≥ 2, n ≥ 3. M2(m, n) is a spanning subgraph of Pm × Cn which has interesting fault hamiltonian properties. We show that M2(m, n) with odd n is hamiltonian-connected and 1-fault hamiltonian. For even n, M2(m, n), which is bipartite, with a single faulty element is shown to be 1-fault strongly hamiltonian-laceable. In previous works[1, 2], it was shown that Pm×Cn also has these hamiltonian properties. Our result shows that two additional wraparound edges are sufficient for an m × n mesh to have such properties rather than m wraparound edges. As an application of fault-hamiltonicity of M2(m, n), we show that the n-dimensional hypercube is strongly hamiltonian laceable if there are at most n− 2 faulty elements and at most one faulty vertex.