Hypercubes are famous interconnection networks. The shuffle cubes are proposed as its variation with lower diameter. In this paper, we concentrate on the fault tolerant hamiltonicity and hamiltonian connectivity of the shuffle cube. We say that a graph G is f-hamiltonian if there is a hamiltonian cycle in G − F and is f-hamiltonian connected if there is a hamiltonian path between any two vertic...

To embed a ring in a hypercube is to find a Hamiltonian cycle through every node of the hypercube. It is obvious that no 2 n node Hamiltonian cycle exists in an n-dimensional faulty hypercube which has at least one faulty node. However, if a hypercube has faulty links only and the number of faulty links is at most n − 2, at least one 2 n -node Hamiltonian cycle can be found. In this paper, we p...

by Derek Holton and Robert E. L. Aldred Department of Mathematics and Statistics University of Otago This paper seeks to review some ideas and results relating to Hamiltonian graphs. We list the well known results which are to be found in most undergraduate graph theory courses and then consider some old theorems which are fundamental to planar graphs. By restricting our attention to 3-connecte...

We show how to find a Hamiltonian cycle in a graph of degree at most three with n vertices, in time O(2) ≈ 1.260 and linear space. Our algorithm can find the minimum weight Hamiltonian cycle (traveling salesman problem), in the same time bound. We can also count or list all Hamiltonian cycles in a degree three graph in time O(2) ≈ 1.297. We also solve the traveling salesman problem in graphs of...

Let R and F be two disjoint edge sets in an n-dimensional hypercube Qn. We give two constructing methods to build a Hamiltonian cycle or path that includes all the edges of R but excludes all of F . Besides, considering every vertex of Qn incident to at most n − 2 edges of F , we show that a Hamiltonian cycle exists if (A) |R| + 2|F | ≤ 2n− 3 when |R| ≥ 2, or (B) |R| + 2|F | ≤ 4n− 9 when |R| ≤ ...

Consider a random graph G composed of a Hamiltonian cycle on n labeled vertices and dn random edges that “hide” the cycle. Is it possible to unravel the structure, that is, to efficiently find a Hamiltonian cycle in G? We describe an O(n3 logn) steps algorithm A for this purpose, and prove that it succeeds almost surely. Part one of A properly covers the “trouble spots” of G by a collection of ...

A power assignment is an assignment of transmission power to each of the wireless nodes of a wireless network, so that the induced graph satisfies some desired properties. The cost of a power assignment is the sum of the assigned powers. In this paper, we consider the dual power assignment problem, in which each wireless node is assigned a highor low-power level, so that the induced graph is st...

In this paper we consider an approach to solve the Hamiltonian strictly alternating cycle problem. This approach is based on constructing logical models for the problem.

A nonempty vertex set X ⊆ V (G) of a hamiltonian graph G is called an H-force set of G if every X-cycle of G (i.e. a cycle of G containing all vertices of X) is hamiltonian. The H-force number h(G) of a graph G is defined to be the smallest cardinality of an H-force set of G. In the paper the study of this parameter is introduced and its value or a lower bound for outerplanar graphs, planar gra...

We show that the Hamiltonian cycle problem is NP-complete for a class of planar graphs named triangle graphs that are closely related to inner-triangulated graphs. We present a linear-time heuristic algorithm that nds a solution at most one-third longer than the optimum solution and use it to obtain a fast rendering algorithm on triangular meshes in computer graphics.