Hamiltonian paths on directed grids

Journal of Integer Sequences

Given a grid graph G of size mn, we study the number i(m, n) of independent sets in G, as well as b(m, n), the number of maximal such sets. It turns out that the initial cases b(1, n) and b(2, n) lead to a Padovan and a Fibonacci sequence. To determine b(m, n) for m > 2 we present an adaptation of the transfer matrix method, well known for calculating i(m, n). Finally, we apply our method to obtain explicit values of b(m, n) for m = 3, 4, 5 and provide the corresponding generating functions.

Discrete Mathematics, 1990

arXiv (Cornell University), 2016

We study the Hamiltonian path problem in C−shaped grid graphs, and present the necessary and sufficient conditions for the existence of a Hamiltonian path between two given vertices in these graphs. We also give a linear-time algorithm for finding a Hamiltonian path between two given vertices of a C−shaped grid graph, if it exists.

Lecture Notes in Computer Science, 2022

A simple s, t path P in a rectangular grid graph G is a Hamiltonian path from the top-left corner s to the bottom-right corner t such that each internal subpath of P with both endpoints a and b on the boundary of G has the minimum number of bends needed to travel from a to b (i.e., 0, 1, or 2 bends, depending on whether a and b are on opposite, adjacent, or the same side of the bounding rectangle). Here, we show that P can be reconfigured to any other simple s, t path of G by switching 2 × 2 squares, where at most 5|G|/4 such operations are required. Furthermore, each square-switch is done in O(1) time and keeps the resulting path in the same family of simple s, t paths. Our reconfiguration result proves that the Hamiltonian path graph G for simple s, t paths is connected and has diameter at most 5|G|/4 which is asymptotically tight.

Proceedings of the 15th Iapr International Conference on Discrete Geometry For Computer Imagery, 2009

Among the polyominoes that tile the plane by translation, the so-called squares have been conjectured to tile the plane in at most two distinct ways (these are called double squares). In this paper, we study two families of tiles : one is directly linked to Christoffel words while the other stems from the Fibonacci sequence. We show that these polyominoes are double squares, revealing strong connections between discrete geometry and other areas by means of combinatorics on words.

2020

A grid graph is a finite embedded subgraph of the infinite integer grid. A solid grid graph is a grid graph without holes, i.e., each bounded face of the graph is a unit square. The reconfiguration problem for Hamiltonian cycle or path in a sold grid graph G asks the following question: given two Hamiltonian cycles (or paths) of G, can we transform one cycle (or path) to the other using some “operation” such that we get a Hamiltonian cycle (or path) of G in the intermediate steps (i.e., after each application of the operation)? In this thesis, we investigate reconfiguration problems for Hamiltonian cycles and paths in the context of two types of solid graphs: rectangular grid graphs, which have a rectangular outer boundary, and L-shaped grid graphs, which have a single reflex corner on the outer boundary, under three operations we define, flip, transpose and switch, that are local in the grid. Reconfiguration of Hamiltonian cycles and paths in embedded grid graphs has potential appl...

In polymer science, Hamiltonian paths and Hamiltonian circuits can serve as excellent simple models for dense packed globular proteins. Generation and enumeration of Hamiltonian paths and Hamiltonian circuits (compact conformations of a chain) are needed to investigate thermodynamics of protein folding. Hamiltonian circuits are a mathematical idealization of polymer melts, too. The number of Hamiltonian cycles on a graph corresponds to the entropy of a polymer system. In this paper, we present new characterizations of the Hamiltonian cycles in a labeled rectangular grid graph P m × P n and in a labeled thin cylinder grid graph C m ×P n . We proved that for any fixed m, the numbers of Hamiltonian cycles in these grid graphs, as sequences with counter n, are determined by linear recurrences. The computational method outlined here for finding these difference equations together with the initial terms of the sequences has been implemented. The generating functions of the sequences are given explicitly for some values of m. The obtained data are consistent with data obtained in the works by Kloczkowski and Jernigan, and Schmalz et al.

arXiv: Combinatorics, 2018

We introduce a new type of lattice path, called brick-wall lattice path, and we derive a formula which counts the number of paths on these lattices imposing certain restrictions on the Cartesian plane. Connections to the Fibonacci sequence, as well as to other sequences of numbers, are given. Finally, we use these brick-wall lattice paths to determine the first two non-zero coefficients of the reliability polynomials associated with particular two-terminal networks known as hammocks.

2008

The interconnection network is an essential component of a distributed system or of a supercomputer based on large-sale parallel processing. Because in distributed systems the communication between processors is based on message exchange, the network topology is of a great importance. The interconnection network can be seen as a graph and the properties of a network can be studied using combinatorics and graph theory. A number of interconnection network topologies have been studied. The Extended Fibonacci Cube, EFC, is a topology which provides good properties for an interconnection network regarding diameter, node degree, recursive decomposition, embeddability and communication algorithms. In this research we present some properties of the Extended Fibonacci Cubes, we define a Gray code for extended Fibonacci cubes and show how a hamiltonian path, a hamiltonian cycle and a 2D mesh can be embedded in an Extended Fibonacci Cube.

2019

For a (p, q)-graph G, if the vertices of G can be arranged in a sequence v1, v2, . . . , vp such that for each i = 1, 2, . . . , p − 1, the distance from vi to vi+1 equal to k, then the sequence is called an AL(k)-step traversal. Furthermore, if d(vp, v1) = k, the sequence v1, v2, . . . , vp, v1 is called a k-step Hamiltonian tour and G is k-step Hamiltonian. In this paper we completely determine which rectangular grid graphs are 3-step Hamiltonian and show that the torus graph Cm×Cn is 3-step Hamiltonian for all m ≥ 3, n ≥ 5.