Narrow Results

Low-dimensional Tori are regularly used as interconnection networks in distributed-memory parallel computers. This paper investigates the fault-Hamiltonicity of two-dimensional Tori. A sufficient condition is derived for the graph Row-Torus(m, 2n + 1) with two faulty edges to have a Hamiltonian cycle, where m ≥ 3 and n ≥ 1. By applying the fault-Hamiltonicity of Row-Torus to a two-dimensional torus, we show that Torus(m, n), m, n ≥ 5, with at most four faulty edges is Hamiltonian if the following two conditions are satisfied: (1) the degree of every vertex is at least two, and (2) there do not exist a pair of nonadjacent vertices in a 4-cycle whose degrees are both two after faulty edges are removed.

In this paper, we study the fault-tolerant capability of hypercubes with respect to the hamiltonian property based on the concept of forbidden faulty sets. We show, with the assumption that each vertex is incident with at least three fault-free edges, that an $n$-dimensional hypercube contains a fault-free hamiltonian cycle, even if there are up to $(4n-13)$ edge faults. Moreover, we give an example to show that the result is optimal with respect to the number of edge faults tolerated.

Let G₁ and G₂ be two planar graphs having some vertices in common. A simultaneous embedding of G₁ and G₂ is a pair of crossing-free drawings of G₁ and G₂ such that each vertex in common is represented by the same point in both drawings. In this paper we show that an outerplanar graph and a simple path can be simultaneously embedded with fixed edges such that the edges in common are straight-line segments while the other edges of the outerplanar graph can have at most one bend per edge. We then exploit the technique for outerplanar graphs and paths to study simultaneous embeddings of other pairs of graphs. Namely, we study simultaneous embedding with fixed edges of: (i) two outerplanar graphs sharing a forest of paths and (ii) an outerplanar graph and a cycle.