Hamiltonian Cycle Research Articles

An asymmetric traveling salesman problem based matheuristic algorithm for flowshop group scheduling problem

The flowshop group scheduling problem (FGSP) has become a hot research problem owing to its practical applications in modern industry in recent years. The FGSP can be regarded as a combination of two coupled sub-problems. One is the group scheduling sub-problem with sequence-dependent setup times. The other is the job scheduling sub-problem within each group. A mixed integer linear programming model is built for the FGSP with the makespan criterion. Based on the problem-specific knowledge, i.e., the sequence-dependent group setup times are greater than the processing time of jobs, and the number of machines is small, the group scheduling sub-problem is approximated into an asymmetric traveling salesman problem (ATSP). Then, a matheuristic algorithm (MA) is proposed by integrating a branch-and-cut algorithm and an iterated greedy (IG) algorithm, where the branch-and-cut algorithm is used to generate the optimal Hamiltonian circuit for sub-group sequences of a group sequence obtained by the IG. On 405 test instances, the proposed MA performs significantly better than several state-of-the-art algorithms in the literature.

CMSTR: A Constrained Minimum Spanning Tree Based Routing Protocol for Wireless Sensor Networks

How to extend the network lifetime with given limited energy budget is always one of the main concerns in Wireless Sensor Networks (WSNs). However, imbalanced energy consumption and overlong intra-cluster communication paths are prevalent in the hierarchical routing protocols, which shortens the network lifetime inevitably. To this end, an energy-efficient routing Protocol based on Constrained Minimum Spanning Tree (CMSTR) is proposed in this paper. To be specific, a new multichain routing scheme to balance the energy consumption for intra-cluster communications is presented. Based on the multichain routing scheme, the problem of establishing intra-cluster routing is transformed into a shortest Hamiltonian path problem on the basis of a graph-theoretic analysis model, which is solved through a Constrained Minimum Spanning Tree (CMST) algorithm proposed in this paper, with the aim to obtain the initial path for intra-cluster communications. In order to shorten the initial path length to obtain higher energy-efficient chain routes, a Neighbor Node Replacement (NNR) algorithm and a Link Intersection Detection and Elimination (LIDE) algorithm are proposed, in which the problem of potential long links and intersections is to be effectively alleviated. With shorter chain routes, unnecessary intra-cluster communication energy depletion can be reduced accordingly. In order to evaluate the performance of CMSTR, extensive simulation experiments are conducted. The results show that CMSTR can greatly prolong the network lifetime with regard to the metrics of FND and HND. To be specific, compared with LEACH, R-LEACH, and DCMSTR, the value of FND increased by 800%, 540% and 57%, that of HND increased by 322%, 286% and 22%, and overall network lifetime (AND) increased by 29%, 10% and 5%, respectively. Besides, CMSTR has a stable and lowest packet loss percentage (0.4%). In summary, CMSTR has excellent performance in terms of energy efficiency and network stability.

Heavy and light paths and Hamilton cycles

Given a graph G, we denote by f(G,u0,k) the number of paths of length k in G starting from u0. In graphs of maximum degree 3, with edge weights i.i.d. with exp(1), we provide a simple proof showing that (under the assumption that f(G,u0,k)=ω(1)) the expected weight of the heaviest path of length k in G starting from u0 is at least(1−o(1))(k+log2⁡(f(G,u0,k))2), and the expected weight of the lightest path of length k in G starting from u0 is at most(1+o(1))(k−log2⁡(f(G,u0,k))2).We demonstrate the immediate implication of this result for Hamilton paths and Hamilton cycles in random cubic graphs, where we show that typically there exist paths and cycles of such weight as well. Finally, we discuss the connection of this result to the question of a longest cycle in the giant component of supercritical G(n,p).

Novel schemes for embedding Hamiltonian paths and cycles in balanced hypercubes with exponential faulty edges

The balanced hypercube BHn plays an essential role in large-scale parallel and distributed computing systems. With the increasing probability of edge faults in large-scale networks and the widespread applications of Hamiltonian paths and cycles, it is especially essential to study the fault tolerance of networks in the presence of Hamiltonian paths and cycles. However, existing researches on edge faults ignore that it is almost impossible for all faulty edges to be concentrated in a certain dimension. Thus, the fault tolerance performance of interconnection networks is severely underestimated. This paper focuses on three measures, t-partition-edge fault-tolerant Hamiltonian, t-partition-edge fault-tolerant Hamiltonian laceable, and t-partition-edge fault-tolerant strongly Hamiltonian laceable, and utilizes these measures to explore the existence of Hamiltonian paths and cycles in balanced hypercubes with exponentially faulty edges. We show that the BHn is 2n−1-partition-edge fault-tolerant Hamiltonian laceable, 2n−1-partition-edge fault-tolerant Hamiltonian, and (2n−1−1)-partition-edge fault-tolerant strongly Hamiltonian laceable for n≥2. Comparison results show the partitioned fault model can provide the exponential fault tolerance as the value of the dimension n grows.

Application of Graph Theory in Real Life to Develop Routes

Graph theory is a special part of discrete mathematics, in which we describe the relationship between points and lines. It plays a significant role in the field of science, engineering and technology. With the help of Graph theory, we plan to develop bus routes to pick up students and take them to school. We denote each stop as a vertex, and the route as an edge. In this research paper, we use the Hamiltonian path to represent the efficiency of including each vertex within the route. Whether visiting a water park, theme park, or zoo, we plan an efficient route to visit a selected attraction or all of the attractions. Each vertex within the graph is represented by a Hamiltonian path or circuit. The obtained result is efficient to elucidate the methodology.

Properly Colored Hamilton Cycles in Dirac-Type Hypergraphs

We consider a robust variant of Dirac-type problems in $k$-uniform hypergraphs. For instance, we prove that if $\mathcal{H}$ is a $k$-uniform hypergraph with minimum codegree at least $\left(\frac 12 + \gamma \right)n$, $\gamma >0$, and $n$ is sufficiently large, then any edge coloring $\phi$ satisfying appropriate local constraints yields a properly colored tight Hamilton cycle in $\mathcal{H}$. Similar results for loose cycles are also shown.

A perturbation based modified BAT algorithm for priority based covering salesman problem

In this study, Bat algorithm (BA) is modified along with K-opt operation and one newly proposed perturbation approach to solve the well known covering salesman problem (CSP). Here, along with the restriction of the radial distances of the unvisited cities from the visited cities another restriction is imposed where a priority is given to some cities for the inclusion in the tour, i.e., some clusters to be created where the prioritised cities must be the visiting cities and the corresponding CSP is named as Prioritised CSP (PCSP). In the algorithm, 3-opt and 4-opt operations are used for two different purposes. The 4-opt operation is applied for generating an initial solution set of CSP for the BA and the 3-opt operation generates some perturbed solutions of a solution. A new perturbation approach is proposed for generating neighbour solutions of a potential solution where the exchange of some cities in the tour is made and is named as K-bit exchange operation. The proposed solution approach for the CSP and PCSP is named as the modified BA embedded with K-bit exchange and K-opt operation (MBAKEKO). It is a two-stage algorithm where in the first stage of the algorithm the clustering of the cities is done with respect to a fixed visiting city of each cluster in such a manner that the distances of the other cities of the cluster must lie with in the fixed covering distance of the problem and in the second stage the BA is applied to find the minimum cost Hamiltonian circuit by passing through the visiting cities of the clusters. MBAKEKO is tested with a set of benchmark test problems with significantly large sizes from the TSPLIB. To measure the performance of MBAKEKO, its results are compared with the results of different well-known approaches for CSPs available in the literature. It is observed from the comparison studies that MBAKEKO searches the minimum cost tour for any of the considered instances compared to all other well-known algorithms in the literature. It can be concluded from the numerical studies that the performance of MBAKEKO is better with respect to the state-of-the-art algorithms available in the literature.

Large Sets of Hamilton Cycle Decompositions of a Class of Uniform Hypergraphs

We use the Katona-Kierstead definition of a Hamilton cycle in a uniform hypergraph. We construct a polynomial to find the Hamilton cycle decomposition. Then the existence of large sets of these decompositions is given by using a complete automorphism group.

On the number of perfect matchings in the line graph of a traceable graph

On the number of perfect matchings in the line graph of a traceable graph

Certifying Fully Dynamic Algorithms for Recognition and Hamiltonicity of Threshold and Chain Graphs

Solving problems on graphs dynamically calls for algorithms to function under repeated modifications to the graph and to be more efficient than solving the problem for the whole graph from scratch after each modification. Dynamic algorithms have been considered for several graph properties, for example connectivity, shortest paths and graph recognition. In this paper we present fully dynamic algorithms for the recognition of threshold graphs and chain graphs, which are optimal in the sense that the costs per modification are linear in the number of modified edges. Furthermore, our algorithms also consider the addition and deletion of sets of vertices as well as edges. In the negative case, i.e., where the graph is not a threshold graph or chain graph anymore, our algorithms return a certificate of constant size. Additionally, we present optimal fully dynamic algorithms for the Hamiltonian cycle problem and the Hamiltonian path problem on threshold and chain graphs which return a vertex cutset as certificate for the non-existence of such a path or cycle in the negative case.

Oriented discrepancy of Hamilton cycles

AbstractWe propose the following extension of Dirac's theorem: if is a graph with vertices and minimum degree , then in every orientation of there is a Hamilton cycle with at least edges oriented in the same direction. We prove an approximate version of this conjecture, showing that minimum degree guarantees a Hamilton cycle with at least edges oriented in the same direction. We also study the analogous problem for random graphs, showing that if the edge probability is above the Hamiltonicity threshold, then, with high probability, in every orientation of there is a Hamilton cycle with edges oriented in the same direction.

Toughness, hamiltonicity and spectral radius in graphs

The study of the existence of Hamiltonian cycles in a graph is a classical problem in graph theory. By incorporating toughness and spectral conditions, we can consider Chvátal’s conjecture from another perspective: What is the spectral condition to guarantee the existence of a Hamiltonian cycle among t-tough graphs? We first give the answer to 1-tough graphs, i.e. if ρ(G)≥ρ(Mn), then G contains a Hamiltonian cycle, unless G≅Mn, where Mn=K1∇Kn−4+3 and Kn−4+3 is the graph obtained from 3K1∪Kn−4 by adding three independent edges between 3K1 and Kn−4. The Brouwer’s toughness theorem states that every d-regular connected graph always has t(G)>dλ−1 where λ is the second largest absolute eigenvalue of the adjacency matrix. In this paper, we extend the result in terms of its spectral radius, i.e. we provide a spectral condition for a graph to be 1-tough with minimum degree δ and to be t-tough, respectively.

On Hamiltonian property of bi-Cayley graphs

ABSTRACT For a finite group G, and a subset S of G (possibly, contains the identity element), the bi-Cayley graph Bcay(G, S) of G with respect to S is defined as the undirected bipartite graph with vertex set and edge set . In this paper, we give some sufficient conditions for the existence of hamiltonian cycle in , where is a semiproduct of Zm by a subgroup H of G. In addition, we introduce a new graph operation, which produces some classes of bi-Cayley graphs having hamiltonian cycle.

Special Graphs of Euler’s Family* and Tracing Algorithm- (A New Approach)

There are in graph theory, some known graphs which date back from centuries. [Euler graph, Hamiltonian graph etc.] These graphs are basic roots for development of graph theory. In this paper we have discussed the novel concept of tracing Euler tour. It depends on the concept of Link vertex - a join vertex of finite number of cycles as components of Euler graph. In addition to this, a new notion of isomorphic transformation of given graph on to a given line segment known as ‘Linear Graph’ also plays an important role for tracing the Euler graph.

Modification in Two-Connected Graph with Gallai’s Property in 2-Dimensional and 3-Dimensional Graph Containing 19 Vertices

The graph theory plays an important role in the network analysis, social networking as well as in many engineering fields such as electrical circuits, artificial intelligence, architecture, making the design or pattern of roads, buildings, shopping mall and etc. Due to this wide range application human enjoying her life with peacefully, Graph theory creates a way for human being to connect among themselves by social network. All above applications based on graph or molecule which may be the planer, non-planer and Peterson graph or etc. Peterson graph is the most important and reasonable example of Hypo-Hamiltonian graph. In the earlier, it was found as a hypo-traceable graph (graph which has not Hamiltonian graph. Naeem et al has worked on “A Two-Connected Graph with Gallai’s Property” In his research paper he has applied the property and has found the longest path and cycle in the graph. In this research paper we will develop the 3-dimensional graph of computational molecule contains 19 vertices and will split it into three different planes (xy, xz and yz-plane), and will find the longest path, longest cycle the molecule. The designed graphs can be useful in various fields of science and technology including computational geometry, networking, theoretical computer science and circuit designing.

A Unified Grid Approach Using Hamiltonian Paths for Computing Aerodynamic Flows

A solution algorithm using Hamiltonian paths is presented as a unified grid approach for rotorcraft applications. Hidden line structures are robustly identified on general two- and three-dimensional unstructured grids with mixed elements, providing a framework for line-based solvers. A pure quadrilateral/hexahedral mesh is a prerequisite for line identification and enables approximate factorisation along the lines. The numerical efficiency obtained using the line-implicit method on various unstructured grids is better than that of the point-implicit method. Both finite-difference and gradient reconstructions are possible regardless of grid type. A combined reconstruction method is applied, which uses different reconstructions simultaneously but for different grid directions. Finally, the solution convergence rate is further improved using a preconditioned generalized minimal residual method (GMRES), where the preconditioning step is performed using the efficient line-implicit method.

Analysis of factors affecting scientific migration move and distance by academic age, migrant type, and country: Migrant researchers in the field of business and management

This study examines the relationship between migration move and migration distance among migrant researchers on the basis of academic age, migrant type, and country; it also explores migration across continents and regions. The bibliographic data of 916 migrant researchers in the field of business/management in China, Japan, South Korea, and Taiwan are collected from the Web of Science; Scopus author IDs are used to disambiguate the cross-database data. The Hamiltonian path is used to process the bibliographic data, construct the migration trajectories of individual authors across publications, and identify their locations. The overall results reveal that the three variables (i.e., academic age, migrant type, and country) have significant effects on migration moves. The main migration period for researchers is between the academic age of 3 and 13 years, then the migration rates decrease after the age of 14 years. Taiwan has a higher migration rate (27.15%) than China, Japan, and South Korea. The migration distance of migrant researchers is similar across academic age groups that are at the start of their careers. We further discovered that both cross-continental migration rate and cross-regional migration rate are low (<60%), indicating that researchers are less likely to perform long-distance migrations.

Ramsey theory and thermodynamics

Re-shaping of thermodynamics with the graph theory and Ramsey theory is suggested. Maps built of thermodynamic states are addressed. Thermodynamic states may be attainable and non-attainable by the thermodynamic process in the system of constant mass. We address the following question how large should be a graph describing connections between discrete thermodynamic states to guarantee the appearance of thermodynamic cycles? The Ramsey theory supplies the answer to this question. Direct graphs emerging from the chains of irreversible thermodynamic processes are considered. In any complete directed graph, representing the thermodynamic states of the system the Hamiltonian path is found. Transitive thermodynamic tournaments are addressed. The entire transitive thermodynamic tournament built of irreversible processes does not contain a cycle of length 3, or in other words, the transitive thermodynamic tournament is acyclic and contains no directed thermodynamic cycles.

Hamilton cycles in primitive graphs of order 2rs

After long term efforts, it was recently proved by Du, Kutnar and Marušič in 2021 that except for the Petersen graph, every connected vertex-transitive graph of order rs has a Hamilton cycle, where r and s are primes. A natural topic is to solve the hamiltonian problem for connected vertex-transitive graphs of 2rs. This topic is quite nontrivial, as the problem is still unsolved even for that of r = 5. In this paper, it is shown that except for the Coxeter graph, every connected vertex-transitive graph of order 2rs contains a Hamilton cycle, provided the automorphism group acts primitively on vertices.

The Genus of a Graph: A Survey

The problem of determining the genus for a graph can be dated to the Map Color Conjecture proposed by Heawood in 1890. This was implied to be a Thread Problem by Hilbert and Cohn-Vossen. The conjecture was finally established by Ringel, Youngs, and many other mathematicians. Subsequently, the genera of some special graphs with symmetry were determined. The study of genus embeddings of graphs is closely related to other invariants of a graph. Specifically, the computational complexity is dependent on the genus of the underlying graph for certain well-known NP-hard problems. In this survey, main construction techniques and certain criteria are stated in the topic of the genus of a graph. Most graphs with a known genus are listed. A new theorem is shown that the method of joint trees of a graph is reasonable. Moreover, a formal set is introduced, and related results are obtained. Although a cubic graph of Hamilton cycle is asymmetric, it is interesting that a set of associate surfaces of all its joint trees is a formal set with symmetry.