Hamiltonian Cycle Research Articles

Optimal spread for spanning subgraphs of Dirac hypergraphs

Let G and H be hypergraphs on n vertices, and suppose H has large enough minimum degree to necessarily contain a copy of G as a subgraph. We give a general method to randomly embed G into H with good “spread”. More precisely, for a wide class of G, we find a randomised embedding f:G↪H with the following property: for every s, for any partial embedding f′ of s vertices of G into H, the probability that f extends f′ is at most O(1/n)s. This is a common generalisation of several streams of research surrounding the classical Dirac-type problem.For example, setting s=n, we obtain an asymptotically tight lower bound on the number of embeddings of G into H. This recovers and extends recent results of Glock, Gould, Joos, Kühn, and Osthus and of Montgomery and Pavez-Signé regarding enumerating Hamilton cycles in Dirac hypergraphs. Moreover, using the recent developments surrounding the Kahn–Kalai conjecture, this result implies that many Dirac-type results hold robustly, meaning G still embeds into H after a random sparsification of its edge set. This allows us to recover a recent result of Kang, Kelly, Kühn, Osthus, and Pfenninger and of Pham, Sah, Sawhney, and Simkin for perfect matchings, and obtain novel results for Hamilton cycles and factors in Dirac hypergraphs.Notably, our randomised embedding algorithm is self-contained and does not require Szemerédi's regularity lemma or iterative absorption.

On the Biplanarity of Blowups

The 2-blowup of a graph is obtained by replacing each vertex with two non-adjacent copies; a graph is biplanar if it is the union of two planar graphs. We disprove a conjecture of Gethner that 2-blowups of planar graphs are biplanar: iterated Kleetopes are counterexamples. Additionally, we construct biplanar drawings of 2-blowups of planar graphs whose duals have two-path induced path partitions, and drawings with split thickness two of 2-blowups of 3-chromatic planar graphs, and of graphs that can be decomposed into a Hamiltonian path and a dual Hamiltonian path.

Combining Genetic Algorithm with Local Search Method in Solving Optimization Problems

This research is focused on evolutionary algorithms, with genetic and memetic algorithms discussed in more detail. A graph theory problem related to finding a minimal Hamiltonian cycle in a complete undirected graph (Travelling Salesman Problem—TSP) is considered. The implementations of two approximate algorithms for solving this problem, genetic and memetic, are presented. The main objective of this study is to determine the influence of the local search method versus the influence of the genetic crossover operator on the quality of the solutions generated by the memetic algorithm for the same input data. The results show that when the number of possible Hamiltonian cycles in a graph is increased, the memetic algorithm finds better solutions. The execution time of both algorithms is comparable. Also, the number of solutions that mutated during the execution of the genetic algorithm exceeds 50% of the total number of all solutions generated by the crossover operator. In the memetic algorithm, the number of solutions that mutate does not exceed 10% of the total number of all solutions generated by the crossover operator, summed with those of the local search method.

The hamiltonian path graph is connected for simple s, t paths in rectangular grid graphs

The hamiltonian path graph is connected for simple s, t paths in rectangular grid graphs

Branch and Price for the Stochastic Traveling Salesman Problem with Generalized Latency

We consider an extension of the symmetric traveling salesman problem with generalized latency that explicitly models uncertainty. The stochastic traveling salesman problem with generalized latency (STSP-GL) aims to choose a subset of nodes of an undirected graph and determines a Hamiltonian tour among those nodes, minimizing an objective function that is a weighted combination of route design and passenger routing costs. These nodes are selected to ensure that a predefined percentage of uncertain passenger demand is served with a given probability. We formulate the STSP-GL as a stochastic program and propose a branch-and-price algorithm for solving its deterministic equivalent. We also develop a local search approach with which we improve the performance of the branch-and-price approach. We assess the efficiency of the proposed methods on a set of instances from the literature. We demonstrate that the proposed methods outperform a known benchmark, improving upper bounds by up to 85% and lower bounds by up to 55%. Finally, we show that solutions of the stochastic model are both more cost-effective and robust than those of the deterministic model. History: This paper has been accepted for the Transportation Science Special Issue on TSL Conference 2023. Funding: This work was supported by the Bundesministerium für Bildung und Forschung [Grant 03ZU1105FA]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2023.0417 .

Hamiltonian Properties of Augmented k-Ary n-Cubes with Faulty Edges

The augmented [Formula: see text]-ary [Formula: see text]-cube [Formula: see text] is a recently proposed interconnection network that combines the structural characteristics of [Formula: see text]-ary [Formula: see text]-cube [Formula: see text] and augmented cube [Formula: see text], with better features. In this paper, we investigate the Hamiltonian properties of augmented [Formula: see text]-ary [Formula: see text]-cubes. We prove that the augmented [Formula: see text]-ary [Formula: see text]-cube with any [Formula: see text] (resp. [Formula: see text]) faulty edges admits a fault-free Hamilton cycle (resp. Hamilton path).

Hamiltonian Cycles for Finite Weyl Groupoids

Hamiltonian Cycles for Finite Weyl Groupoids

Improving Broadcast Efficiency of Mobile Wireless Sensor Network Based on Gaussian Network Hamiltonian Path

Improving Broadcast Efficiency of Mobile Wireless Sensor Network Based on Gaussian Network Hamiltonian Path

On Detour Index of Join of Hamilton-connected Graphs

In this paper we contribute to the literature of computational chemistry by providing exact expressions for the detour index of joins of Hamilton-connected ( H C ) graphs. This improves upon existing results by loosening the requirement of a molecular graph being Hamilton-connected and only requirement certain subgraphs to be Hamilton-connected.

An Alternate Form of Merino-Mǐcka-Mütze’s Solution to a Knuth’s Combinatorial Generation Problem

A modification of Merino-Mǐcka-Mütze’s solution to a combinatorial generation problem of Knuth is proposed in this survey. The resulting alternate form to such solution is compatible with a reinterpretation by the author of a proof of existence of Hamilton cycles in the middle-levels graphs. Such reinterpretation is given in terms of a dihedral quotient graph associated to each middle-levels graph. The vertices of such quotient graph represent Dyck words and their associated ordered trees. Those Dyck words are linearly ordered via a rooted tree that covers all their tight, or irreducible, forms, offering an universal reference point of view to express and integrate the periodic paths, or blocks, whose concatenation leads to Hamilton cycles resulting from the said solution.

Pancyclicity of Hamiltonian graphs

An n -vertex graph is Hamiltonian if it contains a cycle that covers all of its vertices, and it is pancyclic if it contains cycles of all lengths from 3 up to n . In 1972, Erdős conjectured that every Hamiltonian graph with independence number at most k and at least n = \Omega(k^{2}) vertices is pancyclic. We prove this old conjecture in a strong form by showing that if such a graph has n = (2+o(1))k^{2} vertices, it is already pancyclic, and this bound is asymptotically best possible.

On the exponential growth of the number of finite asymmetric Hamiltonian graphs

Combinatorial analysis of various algebraic systems and their properties is an urgent problem of modern algebra. In a number of works, the growth function of algebraic systems has been studied, which, for a given natural n, indicates the number of n-element finite algebraic systems of a given type. In many works, the actual problem of studying various transformations of algebraic systems has also been investigated. In particular, it was investigated how "rich" or, conversely, "poor" the considered algebraic systems are with transformations of one kind or another. The paper studies asymmetric Hamiltonian graphs with only an identical automorphism. The article proves that the exponential growth function of asymmetric Hamiltonian graphs is. Комбинаторный анализ различных алгебраических систем и их свойств является актуальной проблемой современной алгебры. В ряде работ изучалась функция роста алгебраических систем, которая для заданного натурального n указывает число n-элементных конечных алгебраических систем заданного вида. Во многих работах исследовалась также актуальная проблема изучения различных преобразований алгебраических систем. В частности, исследовалось, насколько «богаты» или, наоборот, «бедны» рассматриваемые алгебраические системы преобразованиями того или иного вида. В статье изучаются асимметричные гамильтоновы графы, обладающие только тождественным автоморфизмом. В статье доказано, что является экспоненциальной функция роста асимметричных гамильтоновых графов.

Compatible powers of Hamilton cycles in dense graphs

AbstractThe notion of incompatibility system was first proposed by Krivelevich, Lee and Sudakov to formulate the robustness of Hamiltonicity of Dirac graphs. Given a graph , an incompatibility system over is a family such that for every , is a family of edge pairs with . Moreover, for an integer , we say is ‐bounded if for every vertex and its incident edge , there are at most pairs in containing . Krivelevich, Lee and Sudakov proved that there is an universal constant such that for every Dirac graph and every ‐bounded incompatibility system over , there exists a Hamilton cycle where every pair of adjacent edges of satisfies for . This resolves a conjecture posed by Häggkvist in 1988 and such a Hamilton cycle is called compatible (with respect to ). We study high powers of Hamilton cycles in this context and show that for every and , there exists a constant such that for sufficiently large and every ‐bounded incompatibility system over an ‐vertex graph with , there exists a compatible th power of a Hamilton cycle in . Moreover, we give a ‐bounded construction which has minimum degree and contains no compatible th power of a Hamilton cycle.

On vertex‐transitive graphs with a unique hamiltonian cycle

On vertex‐transitive graphs with a unique hamiltonian cycle

The general first Zagreb index conditions for Hamiltonian and traceable graphs

The general first Zagreb index conditions for Hamiltonian and traceable graphs

Discovering symbolic laws directly from trajectories with hamiltonian graph neural networks

Abstract The time evolution of physical systems is described by differential equations, which depend on abstract quantities like energy and force. Traditionally, these quantities are derived as functionals based on observables such as positions and velocities. Discovering these governing symbolic laws is the key to comprehending the interactions in nature. Here, we present a Hamiltonian graph neural network (Hgnn), a physics-enforced Gnn that learns the dynamics of systems directly from their trajectory. We demonstrate the performance of Hgnn on n − springs, n − pendulums, gravitational systems, and binary Lennard Jones systems; Hgnn learns the dynamics in excellent agreement with the ground truth from small amounts of data. We also evaluate the ability of Hgnn to generalize to larger system sizes, and to a hybrid spring-pendulum system that is a combination of two original systems (spring and pendulum) on which the models are trained independently. Finally, employing symbolic regression on the learned Hgnn, we infer the underlying equations relating to the energy functionals, even for complex systems such as the binary Lennard-Jones liquid. Our framework facilitates the interpretable discovery of interaction laws directly from physical system trajectories. Furthermore, this approach can be extended to other systems with topology-dependent dynamics, such as cells, polydisperse gels, or deformable bodies.

Optimization of temperature uniformity for multi microwave source alternating heating based on superpermutation planning enhanced by Harris Hawks optimization

For the optimization of material temperature distribution during multi-source microwave heating processes, we propose utilizing the Harris Hawks Optimization Algorithm to enhance the alternating heating strategy among multiple waveguides with superpermutations. Firstly, we innovatively introduce the concept of superpermutations to assign character weights, which subsequently correlates the alternating sequences of different waveguides. Besides, we employ Hamiltonian path planning to determine the optimal order of waveguide alternation. Then, we apply the Harris Hawks Optimization Algorithm and improve the algorithm based on microwave heating evaluation criteria to optimize the strategy for waveguide alternation and enhance search performance. At last, experimental validation involves heating SiC materials in a microwave reactor to verify the credibility of physical experiments against the simulation model used. This analysis includes assessing temperature uniformity and energy conversion efficiency during the heating process, thereby demonstrating the effectiveness of the proposed heating strategy. The experimental results indicate that temperature uniformity improved by 37.75% to 80.25%, and energy conversion efficiency increased by 7.72%.

Arc-disjoint hamiltonian paths in Cartesian products of directed cycles

Arc-disjoint hamiltonian paths in Cartesian products of directed cycles

Hamiltonian diversity: effectively measuring molecular diversity by shortest Hamiltonian circuits

In recent years, significant advancements have been made in molecular generation algorithms aimed at facilitating drug development, and molecular diversity holds paramount importance within the realm of molecular generation. Nonetheless, the effective quantification of molecular diversity remains an elusive challenge, as extant metrics exemplified by Richness and Internal Diversity fall short in concurrently encapsulating the two main aspects of such diversity: quantity and dissimilarity. To address this quandary, we propose Hamiltonian diversity, a novel molecular diversity metric predicated upon the shortest Hamiltonian circuit. This metric embodies both aspects of molecular diversity in principle, and we implement its calculation with high efficiency and accuracy. Furthermore, through empirical experiments we demonstrate the high consistency of Hamiltonian diversity with real-world chemical diversity, and substantiate its effects in promoting diversity of molecular generation algorithms. Our implementation of Hamiltonian diversity in Python is available at: https://github.com/HXYfighter/HamDiv.Scientific contributionWe propose a more rational molecular diversity metric for the community of cheminformatics and drug development. This metric can be applied to evaluation of existing molecular generation methods and enhancing drug design algorithms.

Planar doodles: Their properties, codes and classification

In this paper, we present those properties of planar doodles, especially when regarded as four-valent graphs, that enable us to classify them into prime and super-prime doodles by analogy to a knot sum. We describe a method for partially characterizing a doodle diagram by a doodle code that describes the complementary regions of the diagram and use that code to enumerate all possible prime and super-prime doodle diagrams via their dual graph. In addition, we explore the relationship between planar doodles and twin groups, and note that a theorem of Tutte means that super-prime doodles have a Hamiltonian circuit and we give examples of prime doodles without a Hamiltonian circuit.