We develop a combinatorial method to show that the dodecahedron graph has, up to rotation and reflection, a unique Hamiltonian cycle. Platonic graphs with this property are called topologically uniquely Hamiltonian. The same method is used to demonstrate topologically distinct Hamiltonian cycles on the icosahedron graph and to show that a regular graph embeddable on the 2-holed torus is topolog...

A c-edge-colored multigraph has each edge colored with one of the c available colors and no two parallel edges have the same color. A proper Hamiltonian cycle is a cycle containing all the vertices of the multigraph such that no two adjacent edges have the same color. In this work we establish sufficient conditions for a multigraph to have a proper Hamiltonian cycle, depending on several parame...

A supergrid graph is a finite induced subgraph of the infinite graph associated with the two-dimensional supergrid. The supergrid graphs contain grid graphs and triangular grid graphs as subgraphs. The Hamiltonian cycle problem for grid and triangular grid graphs was known to be NP-complete. In the past, we have shown that the Hamiltonian cycle problem for supergrid graphs is also NP-complete. ...

In the study of hamiltonian graphs, many well known results use degree conditions to ensure su1⁄2cient edge density for the existence of a hamiltonian cycle. Recently it was shown that the classic degree conditions of Dirac and Ore actually imply far more than the existence of a hamiltonian cycle in a graph G, but also the existence of a 2-factor with exactly k cycles, where 1U k U jV…G†j 4 . I...

A graph is Hamiltonian if it has a cycle that visits every vertex exactly once; such a cycle is called a Hamiltonian cycle. In general, the problem of finding a Hamiltonian cycle in a given graph is an NP-complete problem and a special case of the traveling salesman problem. It is a problem in combinatorial optimization studied in operations research and theoretical computer science; see [Garey...

In this paper, the longest Hamiltonian cycle problem and the longest Hamiltonian path problem are proved to be NPO-complete. @ 1998 Published by Elsevier Science B.V.

The prism over a graph G is the Cartesian product of with complete K2. A hamiltonian if there exists spanning cycle in G, and prism-hamiltonian hamiltonian. Rosenfeld Barnette (1973) [15] conjectured that every 3-connected planar prism-hamiltonian. We construct counterexample to conjecture.

The n-dimensional hypercube Qn is a graph whose vertex set consists of all binary vectors of length n, two vertices being adjacent whenever the corresponding vectors differ in exactly one coordinate. It is well-known that Qn is hamiltonian for every n ≥ 2 and the research on hamiltonian cycles satisfying certain additional properties has received a considerable attention ([7]). The applications...

A triangular grid graph is a finite induced subgraph of the infinite graph associated with the two-dimensional triangular grid. We show that the problem Hamiltonian Cycle is NP-complete for triangular grid graphs, while a hamiltonian cycle in connected, locally connected triangular grid graph can be found in polynomial time. 2000 Mathematics Subject Classification: 05C38 (05C45, 68Q25).

The H-force number of a hamiltonian graph G is the smallest number k with the property that there exists a set W ⊆ V (G) with |W | = k such that each cycle passing through all vertices of W is a hamiltonian cycle. In this paper, we determine the H-force numbers of generalized dodecahedra.