Hamiltonian Cycle Research Articles

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.

Hamilton Cycle on the Wheel Graph

Hamilton Cycle on the Wheel Graph

ALGORITHM FOR CONSTRUCTING THE TRIPLE UNIT GRAPH OF TYPE II OF RING Z_n USING PYTHON

Let be a commutative ring with as the set of all unit elements in . This paper introduces a new graph associated with the ring , called the triple unit graph of type II, denoted by with the vertex set is − {0,1}. In TU2(R), two distinct vertices, and , are adjacent if there exists with and such that . This paper focuses on the algorithm for constructing using Python. This research uses the literature study research method. The Python programming language can be used to observe the characteristic result of the graph. From the patterns generated by the algorithm, some characteristics of are obtained. For example, if is a prime and , then is a connected graph, a complete graph, a regular graph, and a Hamiltonian graph

The number of Hamiltonian cycles in groups of Symmetry

Groups are algebraic structures in Abstract Algebra comprised of a set of elements with a binary operation that satisfies closure, associativity, identity and invertibility. Cayley graphs serve as a visualization tool for groups, as they are capable of illustrating certain structures and properties of groups geometrically. In particular, each element in a group is assigned to a vertex in Cayley graphs. By the group action of left-multiplication, distinct elements in the generating sets can act on each element to create varied directed edges (Meier[1]). By contrast, the presence of a Hamiltonian cycles within a graph demonstrates its level of connectivity. In this research paper, utilizing directed Cayley graphs, we present a series of conjectures and theorems regarding the number and existence of Hamiltonian cycles within Dihedral groups, Symmetric groups of Platonic solids and Symmetric groups. By exploring the relationship between Abstract groups and Hamiltonian graphs, this work contributes to the broader field of research pertaining to Groups of Symmetry and Geometric Group Theory.

Cycle products and efficient vectors in reciprocal matrices

We focus on the relationship between Hamiltonian cycle products and efficient vectors for a reciprocal matrix $A$, to more deeply understand the latter. This facilitates a new description of the set of efficient vectors (as a union of convex subsets), greater understanding of convexity within this set and of order reversals in efficient vectors. A straightforward description of all efficient vectors for an $n$-by-$n$, column perturbed consistent matrix is given; it is the union of at most $(n-1)(n-2)/2$ convex sets.

Non-crossing Hamiltonian Paths and Cycles in Output-Polynomial Time

We show that, for planar point sets, the number of non-crossing Hamiltonian paths is polynomially bounded in the number of non-crossing paths, and the number of non-crossing Hamiltonian cycles (polygonalizations) is polynomially bounded in the number of surrounding cycles. As a consequence, we can list the non-crossing Hamiltonian paths or the polygonalizations, in time polynomial in the output size, by filtering the output of simple backtracking algorithms for non-crossing paths or surrounding cycles respectively. We do not assume that the points are in general position. To prove these results we relate the numbers of non-crossing structures to two easily-computed parameters of the point set: the minimum number of points whose removal results in a collinear set, and the number of points interior to the convex hull. These relations also lead to polynomial-time approximation algorithms for the numbers of structures of all four types, accurate to within a constant factor of the logarithm of these numbers.

An inverse eigenvalue problem for structured matrices determined by graph pairs

Given a pair of real symmetric matrices A,B∈Rn×n with nonzero patterns determined by the edges of any pair of chosen graphs on n vertices, we consider an inverse eigenvalue problem for the structured matrix C=[ABIO]∈R2n×2n. We conjecture that C can attain any spectrum that is closed under conjugation. We use a structured Jacobian method to prove this conjecture for A and B of orders at most 4 or when the graph of A has a Hamilton path, and prove a weaker version of this conjecture for any pair of graphs with a restriction on the multiplicities of eigenvalues of C.

Hamiltonian Cycles on Ammann-Beenker Tilings

We provide a simple algorithm for constructing Hamiltonian graph cycles (visiting every vertex exactly once) on a set of arbitrarily large finite subgraphs of aperiodic two-dimensional Ammann-Beenker (AB) tilings. Using this result, and the discrete scale symmetry of AB tilings, we find exact solutions to a range of other problems which lie in the complexity class NP-complete for general graphs. These include the equal-weight traveling salesperson problem, providing, for example, the most efficient route a scanning tunneling microscope tip could take to image the atoms of physical quasicrystals with AB symmetries; the longest path problem, whose solution demonstrates that collections of flexible molecules of any length can adsorb onto AB quasicrystal surfaces at density one, with possible applications to catalysis; and the three-coloring problem, giving ground states for the q-state Potts model (q≥3) of magnetic interactions defined on the planar dual to AB, which may provide useful models for protein folding. Published by the American Physical Society 2024

Perceptron model application for traceability risk in spun pile manufacturing

A precise risk assessment in a production line constitutes a significant item to identify susceptible areas where there is a possibility of product quality degradation. This also applies to the precast concrete production line in Indonesia that has a spun pile product. Based on a risk assessment activity conducted in this study, it is proposed to build a traceability model in order to maintain and even improve the spun pile product quality in Indonesia. The approach used was the Neural Network of the perceptron model for weighing and will result in a defined traceability path in the context of reducing defects and even failed spun pile products. The simulation result showed that the model has been able to detect risky path possibilities to reduce product quality. The accumulation result of high-risk and medium-risk paths in this study showed that closer to product finalization, the risk will be higher. It is evident that when assessing Indicators, the order from the highest accumulation value first is Curing & Demolding and Stressing & Spinning at 29% each, Casting at 14%, Forming & Setting at 14%, and lastly Cutting & Heading at 14%. Regarding the risk assessment for activities, the first position is Curing & Demolding and Stressing & Spinning with 30% each, the second is Casting and Forming & Setting with 15% each, and the third is Cutting & Heading with 10%.

Positive vectors, pairwise comparison matrices and directed Hamiltonian cycles

In the Analytic Hierarchy Process (AHP) the efficient vectors for a pairwise comparison matrix (PC matrix) are based on the principle of Pareto optimal decisions. To infer the efficiency of a vector for a PC matrix we construct a directed Hamiltonian cycle of a certain digraph associated with the PC matrix and the vector. We describe advantages of using this process over using the strong connectivity of the digraph. As an application of our process we find efficient vectors for a PC matrix, A, obtained from a consistent one by perturbing three entries above the main diagonal and the corresponding reciprocal entries, in a way that there is a square submatrix of A of order 2 containing three of the perturbed entries and not containing a diagonal entry of A. For completeness, we include examples showing conditions under which, when deleting a certain entry of an efficient vector for the square matrix A of order n, we have a non-efficient vector for the corresponding square principal submatrix of order n-1 of A.

Hamiltonian Laceability of Hypercubes with Prescribed Linear Forest and/or Faulty Edges

A node in the n -dimensional hypercube Q n is called an odd node (resp. an even node) if the sum of all digits of the node is odd (resp. even). Let F ⊂ E ( Q n ) and let L be a linear forest in Q n − F such that | E ( L ) | + | F | ≤ n − 2 for n ≥ 2 . Let x be an odd node and y an even node in Q n such that none of the paths in L has x or y as internal node or both of them as end nodes. In this note, we prove that there is a Hamiltonian path between x and y passing through L in Q n − F . The upper bound n − 2 on | E ( L ) | + | F | is sharp.

Introducing 3-Path Domination in Graphs

The dominating set of a graph G is a set of vertices D such that for every v ∈ V ( G ) either v ∈ D or v is adjacent to a vertex in D . The domination number, denoted γ ( G ) , is the minimum number of vertices in a dominating set. In 1998, Haynes and Slater [1] introduced paired-domination. Building on paired-domination, we introduce 3-path domination. We define a 3-path dominating set of G to be D = { Q 1 , Q 2 , … , Q k | Q i is a 3-path } such that the vertex set V ( D ) = V ( Q 1 ) ∪ V ( Q 2 ) ∪ ⋯ ∪ V ( Q k ) is a dominating set. We define the 3-path domination number, denoted by γ P 3 ( G ) , to be the minimum number of 3-paths needed to dominate G . We show that the 3-path domination problem is NP-complete. We also prove bounds on γ P 3 ( G ) and improve those bounds for particular families of graphs such as Harary graphs, Hamiltonian graphs, and subclasses of trees. In general, we prove γ P 3 ( G ) ≤ n 3 .

Hamiltonian cycles of balanced hypercube with disjoint faulty edges

The balanced hypercube BHn, a variant of the hypercube, is a novel interconnection network topology for massive parallel systems. It is showed in [Theor. Comput. Sci. 947 (2023) 113708] that for any edge subset F of BHn there exists a fault-free Hamiltonian cycle in BHn−F for n≥2 with |F|≤5n−7 if the degree of every vertex in BHn−F is at least two and there exist no f4-cycles in BHn−F. In this paper, we consider the existence of Hamiltonian cycles of BHn when F is a matching (a set of disjoint edges), and show that each edge e∉F lies on a fault-free Hamiltonian cycle of BHn−F with n≥2. The number of faulty edges in F can be up to 22n−1, which is exponential to the dimension n.

Solution of maximum scatter traveling salesman problem through evolutionary approaches

This paper is concerned with a variant of the traveling salesman problem (TSP) called the maximum scatter traveling salesman problem (MSTSP). The goal of MSTSP is to find a Hamiltonian cycle that maximizes the minimum length edge of the cycle. This problem has real-world applications in domains such as manufacturing and medical imaging. Both symmetric and asymmetric versions of the problem are considered in this paper. To address this problem, we have developed two evolutionary approaches. Our first approach is based on a genetic algorithm, whereas the second approach is based on a differential evolution algorithm. Initial solution generation procedure, variation operators (crossover and mutation) used in our approaches are designed considering the characteristics of this problem. The solutions obtained through variation operators are improved further through a local search that is specially adapted for MSTSP. For benchmarking, we have compared the performance of our proposed approaches with the state-of-the-art approaches available in the literature. Our approaches obtained better quality solutions in a shorter time for both symmetric and asymmetric versions of the problem, thereby clearly demonstrating their effectiveness in solving MSTSP.

Spectral radius and rainbow Hamilton paths of a graph

Let G={G1,…,Gn−1} be a family of (not necessarily distinct) graphs on the same vertex set {1,…,n} and λ(Gi) be the spectral radius of the adjacency matrix of Gi. By using the shifting technique, we prove that if λ(Gi)≥n−2 for any 1≤i≤n−1, then G admits a rainbow Hamilton path, i.e., a Hamilton path whose edge comes from different Gi, unless G1=G2=⋯=Gn−1 and G1≃Kn−1∪K1. This result can be viewed as a strengthening of a classical result of Fiedler and Nikiforov (2010) [10]. Moreover, we also provide a spectral condition on a family of graphs to guarantee a rainbow linear forest of given size. Our result extends a recent result of Zhang and Wang (2023) [31].

Tessellation-Based Construction of Air Route for Wireless Sensor Networks Employing UAV.

In this paper, we consider a wireless sensor network consisting of an unmanned aerial vehicle (UAV) acting as a sink node and a number of sensor nodes scattered uncertainly on the ground. In the network, the UAV flies to a spatial point called point of interest and hovers to collect environmental data from neighboring sensor nodes. Then, the UAV proceeds to the next point of interest. The UAV must gather data from all the sensor nodes. On the other hand, a shorter round-trip air route of the UAV is more preferred since a battery-operated UAV needs regular recharging. To satisfy the requirement and to adhere to the recommendation as well, especially in the situation where only vague locational information about sensor nodes is available, we propose a scheme that follows three steps. First, it covers the sensor field of the wireless sensor network with three categories of hexagonal tessellations. Secondly, it establishes a point of interest at the centroid of each tile. Thirdly, it constructs an air route of the UAV, which visits every point of interest along a Hamiltonian cycle on the induced graph. Next, we develop a closed-form expression for the exact flight distance attained by the proposed scheme. For comparative evaluation, we discover some optimal schemes that minimize the flight distance by completely inspecting all patterns and corroborating the property of Hamiltonicity. The flight distance along the air route constructed by the proposed scheme is found to be only slightly longer than the flight distance yielded by an optimal scheme. Furthermore, the proposed scheme is proven to be practically valid when a common multicopter is employed as the sink node.

Nonparametric High-Dimensional Multi-Sample Tests based on Graph Theory

High-dimensional data pose unique challenges for data processing in an era of ever-increasing amounts of data availability. Graph theory can provide a structure of high-dimensional data. We introduce two key properties desirable for graphs in testing homogeneity. Roughly speaking, these properties may be described as: unboundedness of edge counts under the same distribution and boundedness of edge counts under different distributions. It turns out that the minimum spanning tree violates these properties but the shortest Hamiltonian path posses them. Based on the shortest Hamiltonian path, we propose two combinations of edge counts in multiple samples to test for homogeneity. We give the permutation null distributions of proposed statistics when sample sizes go to infinity. The power is analyzed by assuming both sample sizes and dimensionality tend to infinity. Simulations show that our new tests behave very well overall in comparison with various competitors. Real data analysis of tumors and images further convince the value of our proposed tests. Software implementing the test is available in the R package GRelevance. Supplemental materials for this article are available online.

Approximate and Randomized Algorithms for Computing a Second Hamiltonian Cycle

In this paper we consider the following problem: Given a Hamiltonian graph G, and a Hamiltonian cycle C of G, can we compute a second Hamiltonian cycle C′≠C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{\\prime } \ e C$$\\end{document} of G, and if yes, how quickly? If the input graph G satisfies certain conditions (e.g. if every vertex of G is odd, or if the minimum degree is large enough), it is known that such a second Hamiltonian cycle always exists. Despite substantial efforts, no subexponential-time algorithm is known for this problem. In this paper we relax the problem of computing a second Hamiltonian cycle in two ways. First, we consider approximating the length of a second longest cycle on n-vertex graphs with minimum degree δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document} and maximum degree Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta $$\\end{document}. We provide a linear-time algorithm for computing a cycle C′≠C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{\\prime } \ e C$$\\end{document} of length at least n-4α(n+2α)+8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-4\\alpha (\\sqrt{n}+2\\alpha )+8$$\\end{document}, where α=Δ-2δ-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha = \\frac{\\Delta -2}{\\delta -2}$$\\end{document}. This results provides a constructive proof of a recent result by Girão, Kittipassorn, and Narayanan in the regime of Δδ=o(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{\\Delta }{\\delta } = o(\\sqrt{n})$$\\end{document}. Our second relaxation of the problem is probabilistic. We propose a randomized algorithm which computes a second Hamiltonian cycle with high probability, given that the input graph G has a large enough minimum degree. More specifically, we prove that for every 0<p≤0.02\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<p\\le 0.02$$\\end{document}, if the minimum degree of G is at least 8plog8n+4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{8}{p} \\log \\sqrt{8}n + 4$$\\end{document}, then a second Hamiltonian cycle can be computed with probability at least 1-1n50p4+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 - \\frac{1}{n}\\left( \\frac{50}{p^4} + 1 \\right) $$\\end{document} in poly(n)·24pn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$poly(n) \\cdot 2^{4pn}$$\\end{document} time. This result implies that, when the minimum degree δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document} is sufficiently large, we can compute with high probability a second Hamiltonian cycle faster than any known deterministic algorithm. In particular, when δ=ω(logn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta = \\omega (\\log n)$$\\end{document}, our probabilistic algorithm works in 2o(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^{o(n)}$$\\end{document} time.

Cutting Barnette graphs perfectly is hard

A perfect matching cut is a perfect matching that is also a cutset, or equivalently, a perfect matching containing an even number of edges on every cycle. The corresponding algorithmic problem, Perfect Matching Cut, is known to be NP-complete in subcubic bipartite graphs [Le & Telle, TCS '22], but its complexity was open in planar graphs and cubic graphs. We settle both questions simultaneously by showing that Perfect Matching Cut is NP-complete in 3-connected cubic bipartite planar graphs or Barnette graphs. Prior to our work, among problems whose input is solely an undirected graph, only Distance-2 4-Coloring was known to be NP-complete in Barnette graphs. Notably, Hamiltonian Cycle would only join this private club if Barnette's conjecture were refuted.