In this paper we consider the question of determining the maximum number of edges in a hamiltonian graph of order n that contains no 2-factor with more than one cycle, that is, 2-factor hamiltonian graphs. We obtain exact results for both bipartite graphs, and general graphs, and construct extremal graphs in each case. Mathematics Subject Classification (2000). Primary 05C45; Secondary 05C38.

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 ...

A dynamic programming method for enumerating hamiltonian cycles in arbitrary graphs is presented. The method is applied to grid graphs, king’s graphs, triangular grids, and three-dimensional grid graphs, and results are obtained for larger cases than previously published. The approach can easily be modified to enumerate hamiltonian paths and other similar structures.

Let $G$ be a simple graph of order $n$ and let $k$ an integer such that $1\leq k\leq n-1$. The $k$-token $G^{\{k\}}$ is the whose vertices are $k$-subsets $V(G)$, where two adjacent in whenever their symmetric difference pair $G$. In this paper we study Hamiltonicity graphs some join graphs. As consequence, provide infinite family (containing Hamiltonian non-Hamiltonian graphs) for which Hamilt...

We examine the problem of counting the number of Hamiltonian paths and Hamiltonian cycles in outerplanar graphs and planar graphs, respectively. We give an O(nαn) upper bound and an Ω(αn) lower bound on the maximum number of Hamiltonian paths in an outerplanar graph with n vertices, where α ≈ 1.46557 is the unique real root of α = α + 1. For any positive integer n ≥ 6, we define an outerplanar ...

In this paper we extend general grid graphs to the grid graphs consist of polygons tiling on a plane, named polygonal grid graphs. With a cycle basis satisfied polygons tiling, we study the cyclic structure of Hamilton graphs. A Hamilton cycle can be expressed as a symmetric difference of a subset of cycles in the basis. From the combinatorial relations of vertices in the subset of cycles in th...

We use [3] for terminology and notation not defined here and consider finite simple graphs only. The first major result on the existence of hamiltonian cycles in graphs embeddable in surfaces was by H. Whitney [12] in 1931, who proved that 4-connected maximal planar graphs are hamiltonian. In 1956, W.T. Tutte [10,11] generalized Whitney’s result from maximal planar graphs to arbitrary 4-connect...

Let U be the set of cubic planar hamiltonian graphs, A the set of graphs G in U such thatG− v is hamiltonian for any vertex v of G, B the set of graphs G in U such thatG− e is hamiltonian for any edge e of G, and C the set of graphs G in U such that there is a hamiltonian path between any two different vertices of G. With the inclusion and/or exclusion of the sets A,B, and C, U is divided into ...

This paper is a study of the hamiltonicity of proper interval graphs with applications to the guard problem in spiral polygons. We prove that proper interval graphs with ~> 2 vertices have hamiltonian paths, those with ~>3 vertices have hamiltonian cycles, and those with />4 vertices are hamiltonian-connected if and only if they are, respectively, 1-, 2-, or 3-connected. We also study the guard...