There is only one finite, 2-connected, linearly convex graph in the Archimedean triangular tiling that does not have a Hamiltonian cycle. The vertices and polygonal edges of the planar Archimedean tilings 44 and 36 of the plane, partially shown in Figs. 1 and 2, respectively, are called the square tiling graph (STG) and the triangular tiling graph (TTG). (See [1].) A subgraph G of TTG is linear...

We consider the Hamiltonian cycle problem embedded in singularly perturbed (controlled) Markov chains. We also consider a functional on the space of stationary policies of the process that consists of the (1,1)-entry of the fundamental matrices of the Markov chains induced by these policies. We focus on the subset of these policies that induce doubly stochastic probability transition matrices w...

We show the complete graph on n vertices contains a knotted Hamiltonian cycle in every spatial embedding, for n > 7. Moreover, we show that for n > 8, the minimum number of knotted Hamiltonian cycles in every embedding of Kn is at least (n−1)(n−2) . . . (9)(8). We also prove K8 contains at least 3 knotted Hamiltonian cycles in every spatial embedding.

In 1980, Jackson proved that every 2-connected k-regular graph with at most 3k vertices is Hamiltonian. This result has been extended in several papers. In this note, we determine the minimum number of vertices in a connected k-regular graph that is not Hamiltonian, and we also solve the analogous problem for Hamiltonian paths. Further, we characterize the smallest connected k-regular graphs wi...

In this article we consider the problem of finding Hamiltonian cycles on a tetrahedral mesh. A Hamiltonian cycle is a closed loop through a tetrahedral mesh that visits each tetrahedron exactly once. Using techniques of a novel discrete differential geometry of n-simplices, we could immediately obtain Hamiltonian cycles on a rhombic dodecahedronshaped tetrahedral mesh consisting of 24 tetrahedr...

In this paper we characterize those pairs of forbidden subgraphs sufficient to imply various hamiltonian type properties in graphs. In particular, we find all forbidden pairs sufficient, along with a minor connectivity condition, to imply a graph is traceable, hamiltonian, pancyclic, panconnected or cycle extendable. We also consider the case of hamiltonian-connected graphs and present a result...

AgraphG is k-ordered if for every sequence of k distinct vertices ofG, there exists a cycle inG containing these k vertices in the specified order. It is k-ordered-Hamiltonian if, in addition, the required cycle is a Hamiltonian cycle in G. The question of the existence of an infinite class of 3-regular 4-ordered-Hamiltonian graphs was posed in Ng and Schultz in 1997 [2]. At the time, the only ...

An orthogonal double cover (ODC) of the complete graph Kn by a graph G is a collection G of n spanning subgraphs of Kn, all isomorphic toG, such that any two members ofG share exactly one edge and every edge ofKn is contained in exactly two members of G. In the 1980s Hering posed the problem to decide the existence of an ODC for the case that G is an almost-Hamiltonian cycle, i.e. a cycle of le...

Let D be the circulant digraph with n vertices and connection set {2, 3, c}. (Assume D is loopless and has outdegree 3.) Work of S.C. Locke and D.Witte implies that if n is a multiple of 6, c ∈ {(n/2) + 2, (n/2) + 3}, and c is even, then D does not have a hamiltonian cycle. For all other cases, we construct a hamiltonian cycle in D.

We prove that claw-free graphs, containing an induced dominating path, have a Hamiltonian path, and that 2-connected claw-free graphs, containing an induced doubly dominating cycle or a pair of vertices such that there exist two internally disjoint induced dominating paths connecting them, have a Hamiltonian cycle. As a consequence, we obtain linear time algorithms for both problems if the inpu...