Recently there has been interest in DWDM-based optical networks that are assumed to employ exactly two working fibers uniformly on every span (or “link”). At the fiber level such networks are referred to as homogeneous networks. An interesting and highly efficient strategy for protection of such networks is to use a single dark-fiber pcycle formed on a Hamiltonian cycle (if it exists). We show ...

A cycle C of length k is extendable if there is a cycle C′ of length k+1 with V (C) ⊂ V (C′). A graph G = (V,E) of order n is cycle extendable when every cycle C of length k < n is extendable. A chordal graph is a spider intersection graph if it admits an intersection representation which consists of subtrees of a sub-divided star (or spider). In 1990, Hendry conjectured that all hamiltonian ch...

All graphs considered in this paper are finite and undirected, without loops or multiple edges. Let G = (V, E) be a graph. Throughout this paper, let m and n denote the numbers of edges and vertices of graph G, respectively. A connected graph is distance-hereditary if the distance between every two vertices in a connected induced subgraph is the same as that in the original graph. Distance-here...

This paper addresses the existence of Hamiltonian paths and cycles in two-dimensional grids consisting of triangles or quadrilaterals, and three-dimensional grids consisting of tetrahedra or hexahedra. The paths and cycles may be constrained to pass from one element to the next through an edge, through a vertex, or be unconstrained and pass through either. It was previously known that an uncons...

An st-path is a path with the end-vertices s and t. An s-path is a path with an end-vertex s. The results of this paper include necessary and sufficient conditions for a {claw, net}-free graph G with s, t ∈ V (G) and e ∈ E(G) to have (1) a Hamiltonian s-path, (2) a Hamiltonian st-path, (3) a Hamiltonian sand st-paths containing e when G has connectivity one, and (4) a Hamiltonian cycle containi...

An st-path is a path with the end-vertices s and t. An s-path is a path with an end-vertex s. The results of this paper include necessary and sufficient conditions for a {claw, net}-free graph G with s, t ∈ V (G) and e ∈ E(G) to have (1) Hamiltonian s-path, (2) a Hamiltonian st–path, (3) a Hamiltonian sand st-paths containing e when G has connectivity one, and (4) a Hamiltonian cycle containing...

In this paper, we study the parameterized complexity and inapproximability of the Induced Matching problem in hamiltonian bipartite graphs. We show that, given a hamiltonian cycle in a hamiltonian bipartite graph, the problem is W[1]-hard and cannot be solved in time n 1 2 ) unless W[1]=FPT, where n is the number of vertices in the graph. In addition, we show that unless NP=P, the maximum induc...

The hypercube Qn is one of the most well-known and popular interconnection networks and the k-ary n-cubeQn is an enlarged family from Qn that keeps many pleasing properties from hypercubes. In this article, we study the panpositionable hamiltonicity of Qn for k ≥ 3 and n ≥ 2. Let x, y of V (Qn) be two arbitrary vertices and C be a hamiltonian cycle of Qn. We use dC(x, y) to denote the distance ...

Hamiltonian Cycles in Sparse Graphs Alexander Hertel Master of Science Graduate Department of Computer Science University of Toronto 2004 The subject of this thesis is the Hamiltonian Cycle problem, which is of interest in many areas including graph theory, algorithm design, and computational complexity. Named after the famous Irish mathematician Sir William Rowan Hamilton, a Hamiltonian Cycle ...

It is well known that not all 3-connected planar graphs are hamiltonian. Whitney [10] proved that every triangulation of the sphere with no separating triangles is hamiltonian. Tutte [9] proved that every 4-connected planar graph has a Hamilton cycle. Extending Tutte's technique, Thomassen [8] proved that every 4-connected planar graph is in fact Hamilton connected. (A small omission in [8] was...