A Hamiltonian cycle is a closed path through all the vertices of a graph. Since discovering whether a graph has a Hamiltonian path or a Hamiltonian cycle are both NP-complete problems, researchers concentrated on formulating sufficient conditions that ensure Hamiltonicity of a graph. A Information Processing Letters 94(2005), 37-51] presents distance based sufficient conditions for the existenc...

In this paper, we investigate the well-studied Hamiltonian cycle problem, and present an interesting dichotomy result on split graphs. T. Akiyama, T. Nishizeki, and N. Saito [22] have shown that the Hamiltonian cycle problem is NP-complete in planar bipartite graph with maximum degree 3. Using this reduction, we show that the Hamiltonian cycle problem is NP-complete in split graphs. In particul...

In this paper, we first introduce a novel class of graphs, namely supergrid. Supergrid graphs include grid graphs and triangular grid graphs as their subgraphs. The Hamiltonian cycle and path problems for grid graphs and triangular grid graphs were known to be NP-complete. However, they are unknown for supergrid graphs. The Hamiltonian cycle (path) problem on supergrid graphs can be applied to ...

A Hamiltonian cycle is a spanning cycle in a graph, i.e., a cycle through every vertex, and a Hamiltonian path is a spanning path. In this paper we present two theorems stating sufficient conditions for a graph to possess Hamiltonian cycles and Hamiltonian paths. The significance of the theorems is discussed, and it is shown that the famous Ore’s theorem directly follows from our result.  2004...

We describe a general sufficient condition for a Hamiltonian graph to contain another Hamiltonian cycle. We apply it to prove that every longest cycle in a 3-connected cubic graph has a chord. We also verify special cases of an old conjecture of Sheehan on Hamiltonian cycles in 4-regular graphs and a recent conjecture on a second Hamiltonian cycle by Triesch, Nolles, and Vygen. 1997 Academic Press

Fixed-parameter tractability Definition 2. A parameterized problem Π is fixed-parameter tractable (FPT) if there is an algorithm solving Π in time f(k) · poly(n), where n is the instance size, k is the parameter, poly is a polynomial function, and f is a computable function. Theorem 3. Let Π be a decidable parameterized problem. Π has a kernelization ⇔ Π is FPT. 2 A kernel for Hamiltonian Cycle...

An Archimedean graph in R 2 is any finite subgraph of the vertices and edges of one of the 11 Archimedean tilings of R E shown in Figures 1-3. Archimedean filings are commonly characterized (as in Figures 1-3) by the order in which regular polygons (with unit edges) occur around each vertex of the tiling; see Grtinbaum and Shephard [3]. We use the same description for Archimedean subgraphs. Whe...

In this paper, we first introduce a novel class of graphs, namely supergrid. Supergrid graphs include grid graphs and triangular grid graphs as their subgraphs. The Hamiltonian cycle and path problems for grid graphs and triangular grid graphs were known to be NP-complete. However, they are unknown for supergrid graphs. The Hamiltonian cycle (path) problem on supergrid graphs can be applied to ...

In this note we show that deciding the existence of a Hamiltonian cycle in a cubic plane graph is equivalent to the problem of the existence of an associated cubic plane multi-3-gonal graph with a Hamiltonian cycle which takes alternately left and right edges at each successive vertex, i.e. it is also a Petrie cycle. The Petrie Hamiltonian cycle in an n-vertex plane cubic graph can be recognize...

Graphs can be used to model various problems and relations in diverse domains eg. Computer Science, Biological, Chemical and many other. There exist various hard problems in communication networks, which are practically hard but can be shaped in the form of graphs. A Hamiltonian Path is a spanning trail in a network graph i.e. a path through every network node and a spanning cycle in a network ...