This paper describes the study of a special class of 4-regular plane graphs which are Hamiltonian. These graphs are of special interest in knot theory. An algorithm is presented that randomly generates such graphs with n vertices with a fixed (and oriented) Hamiltonian cycle in O(n) time. An exact count of the number of such graphs with n vertices is obtained and the asymptotic growth rate of t...

In this paper we generalize a Theorem of Jung which shows that 1-tough graphs with (G) |V (G)|−4 2 are hamiltonian. Our generalization shows that these graphs contain a wide variety of 2-factors. In fact, these graphs contain not only 2-factors having just one cycle (the hamiltonian case) but 2-factors with k cycles, for any k such that 1 k n−16 4 . © 2004 Elsevier B.V. All rights reserved.

In 1981, Duffus, Gould, and Jacobson showed that every connected graph either has a Hamiltonian path, or contains claw (K1,3) net (a fixed six-vertex graph) as an induced subgraph. This implies subject to being connected, these two are the only minimal (under taking subgraphs) graphs with no path. Brousek (1998) characterized 2-connected, non-Hamiltonian do not contain We characterize 2-connect...

A grid graph is a nite node-induced subgraph of the innnite two-dimensional integer grid. A solid grid graph is a grid graph without holes. For general grid graphs, the Hamiltonian cycle problem is known to be NP-complete. We give a polynomial-time algorithm for the Hamiltonian cycle problem in solid grid graphs, resolving a longstanding open question posed in IPS82]. In fact, our algorithm can...

We survey results and open problems in hamiltonian graph theory centred around three themes: regular graphs, t-tough graphs, and claw-free graphs.

We introduce a new class of graphs which we call P3-dominated graphs. This class properly contains all quasi-claw-free graphs, and hence all claw-free graphs. Let G be a 2-connected P3-dominated graph. We prove that G is hamiltonian if α(G) ≤ κ(G), with two exceptions: K2,3 and K1,1,3. We also prove that G is hamiltonian if G is 3-connected and |V (G)| ≤ 5δ(G) − 5. These results extend known re...

For an integer s ≥ 0, a graph G is s-hamiltonian if for any vertex subset S′ ⊆ V (G) with |S′| ≤ s, G − S′ is hamiltonian. It is well known that if a graph G is s-hamiltonian, then G must be (s + 2)-connected. The converse is not true, as there exist arbitrarily highly connected nonhamiltonian graphs. But for line graphs, we prove that when s ≥ 5, a line graph is s-hamiltonian if and only if it...

Abstract A non-Hermitian Hamiltonian that has an unbroken PT symmetry can be converted by means of a similarity transformation to a physically equivalent Hermitian Hamiltonian. This raises the following question: In which form of the quantum theory, the non-Hermitian or the Hermitian one, is it easier to perform calculations? This paper compares both forms of a non-Hermitian ix quantum-mechanic...

A graph is uniquely hamiltonian if it contains exactly one hamiltonian cycle. In this note we prove that there are no r-regular uniquely hamiltonian graphs when r > 22. This improves upon earlier results of Thomassen.

The Hamiltonian cycle problem is one of the most popular NP-complete problems, and remains NP-complete even if we restrict ourselves to a class of (3-connected cubic) planar graphs [5,9]. Therefore, there seems to be no polynomial-time algorithm for the Hamiltonian cycle problem. However, for certain (nontrivial) classes of restricted graphs, there exist polynomial-time algorithms [3,4,6]. In f...