Hamiltonian Cycle Research Articles

EGS P26 Facilitating the Emergency Surgical Ward Round: The Route to WardNav

Abstract Background The ward round is a corner stone of surgical practice, this is no different when dealing with the acute surgical take. Constant bed pressures result in frequently needing to visit multiple wards, requiring time and effort on staff to ensure patients are reviewed. With frequent rotation of junior doctors, often the responsibility of planning the ward round falls to the ward round leader, who is also making the clinical decisions. We hypothesise a ward round planning tool available to the surgical team would result in more efficient ward rounds, freeing the team to focus on clinical decisions. Methods A survey of junior doctors (foundation doctors and core trainees) was distributed assessing their level of engagement with ward round planning while on surgical rotations at Southmead Hospital, Bristol. An initial planning tool was created in Microsoft Excel utilising a node edge graph and an algorithm based on the Hamiltonian cycle. After further survey feedback, the tool was developed into a web-based app using a version of Djikstra’s algorithm. Results 25 junior doctors responded to the initial survey over two rotations. 100% had to go to multiple wards on a round, 64% actively think about the route required although only 28% would be confident in planning the route. 96% would find a planning tool helpful. Feedback on the Excel based tool from the first cohort revealed appetite for a mobile app-based version. The resultant application was created in R. The final version allows the user to start the round on any ward and creates the optimal route through the building, visually represented on a graph with accompanying text output. Conclusions Junior surgical doctors are not confident with planning ward rounds in a large teaching hospital. An app-based ward navigation tool has the potential to remove ward round planning from daily decision making and engage the junior surgical staff.

A Hamilton Path-Based Approach to DNA Sequence Analysis

The Hamiltonian path is a famous algorithm for determining the existence or nonexistence of a graph. However, for certain types of graphs, we can apply certain conditions to determine its existence. In this paper, we discuss the results show that there can be at most (n-1)/2 edges disjoint Hamiltonian paths in a complete graph. Also, discussed examples of its applications in different fields. Furthermore, the application of Hamilton path in the DNA sequence has been investigated.

On oriented cycles in randomly perturbed digraphs

AbstractIn 2003, Bohman, Frieze, and Martin initiated the study of randomly perturbed graphs and digraphs. For digraphs, they showed that for every $\alpha \gt 0$ , there exists a constant $C$ such that for every $n$ -vertex digraph of minimum semi-degree at least $\alpha n$ , if one adds $Cn$ random edges then asymptotically almost surely the resulting digraph contains a consistently oriented Hamilton cycle. We generalize their result, showing that the hypothesis of this theorem actually asymptotically almost surely ensures the existence of every orientation of a cycle of every possible length, simultaneously. Moreover, we prove that we can relax the minimum semi-degree condition to a minimum total degree condition when considering orientations of a cycle that do not contain a large number of vertices of indegree $1$ . Our proofs make use of a variant of an absorbing method of Montgomery.

A Book Proof of the Middle Levels Theorem

We give a short constructive proof for the existence of a Hamilton cycle in the subgraph of the (2n+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2n+1)$$\\end{document}-dimensional hypercube induced by all vertices with exactly n or n+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n+1$$\\end{document} many 1s.

Rainbow powers of a Hamilton cycle in Gn,p

AbstractWe show that the threshold for having a rainbow copy of a power of a Hamilton cycle in a randomly edge colored copy of is within a constant factor of the uncolored threshold. Our proof requires times the minimum number of colors.

ANALYSIS OF THE DEFERRED SOLUTIONS METHOD FOR FINDING OF HAMILTONIAN CYCLE ON A GRAPH

The subject of research is the solving of the problem of finding a Hamiltonian cycle on a graph, which belongs to the NP complexity class. The purpose of the work is to develop an effective polynomial algorithm for its optimal solving. This work is a continuation of the author's previous work, where the method of deferred solutions is proposed, which uses a iterating over acceptable solutions scheme, which is explaining by the inability to formulate conditions for directly finding the optimal solution. For a graph of n vertices, the size of the iteration space is (n-1)!. For large values n, the time costs are unacceptably large and their reduction is required. During the operation of the deferred solutions method, until the generated solution of the problem becomes a Hamiltonian cycle, it has name – a partial solution. The scheme underlying the deferred solutions method provides: the rejection of the complete construction of all solutions, the simultaneous formation of a set of partial solutions, the discarding of unpromising solutions, the possibility of returning to deferred partial solutions if necessary, the exclusion of the loss of the optimal solution when discarding partial solutions. However, as shown, only when choosing the correct estimate of partial solutions. At first, the real length of the partial solution path was using as an estimate. For an incomplete graph of 20 vertices, the optimal solution found in 0.005 minutes, but on a complete graph of 20 vertices, the search time was commensurate with the search of all possible solutions to the problem. The article analyzes and shows that the real length of the path as an estimate is logically justified and guarantees finding the optimal solution. However, does not always guarantee the minimum time spent searching for it, since when searching through the space of acceptable solutions, a search in width scheme worked out, which entails the construction of almost all acceptable solutions. This explains the different time spent on finding the optimal solution for tests incomplete and complete graphs – the sets of acceptable solutions differ significantly. In this work, as an alternative, another estimate is proposing – the path length of the partial solution, measured in the arcs of the graph. As shown that the use of this estimate leads to a search of solutions in depth. This estimate reduces the time to find a solution, but does not guarantee an optimal result. For the successful application of the method, it is necessary to develop a new estimate of partial solutions that would combine the qualities of the estimates considered.

Correspondence between the Hamiltonian cycle problem and the quantum lattice gauge theory

We propose the correspondence between the Hamiltonian cycle (HC) problem in graph theory and the quantum lattice gauge theory (QZ2LGT) defined on the lattice dual to that graph. For the QZ2LGT, when the coupling parameter g is less than the critical value g c , the ground state is a superposition of all configurations with closed strings of same spins, which can be obtained by using an adiabatic quantum algorithm. A subsequent search for a HC among those closed strings solves the original HC problem. The method is demonstrated for random samples of small graphs.

Embedding Hamiltonian Paths in $k$-Ary $n$-Cubes With Exponentially-Many Faulty Edges

The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -ary <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -cube <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$Q_{n}^{k}$</tex-math></inline-formula> is one of the most popular interconnection networks engaged as the underlying topology of data center networks, on-chip networks, and parallel and distributed systems. Due to the increasing probability of faulty edges in large-scale networks and extensive applications of the Hamiltonian path, it becomes more and more critical to investigate the fault tolerability of interconnection networks when embedding the Hamiltonian path. However, since the existing edge fault models in the current literature only focus on the entire status of faulty edges while ignoring the important information in the edge dimensions, their fault tolerability is narrowed to a minimal scope. This paper first proposes the concept of the partitioned fault model to achieve an exponential scale of fault tolerance. Based on this model, we put forward two novel indicators for the bipartite networks (including <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$Q^{k}_{n}$</tex-math></inline-formula> with even <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> ), named partition-edge fault-tolerant Hamiltonian laceability and partition-edge fault-tolerant hyper-Hamiltonian laceability. Then, we exploit these metrics to explore the existence of Hamiltonian paths and unpaired 2-disjoint path cover in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -ary <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -cubes with large-scale faulty edges. Moreover, we prove that all these results are optimal in the sense that the number of edge faults tolerated has attended to the best upper bound. Our approach is the first time that can still embed a Hamiltonian path and an unpaired 2-disjoint path cover into the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -ary <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -cube even if the faulty edges grow exponentially.

The Hamiltonian Cycle and Travelling Salesperson problems with traversal-dependent edge deletion

Variants of the well-known Hamiltonian Cycle and Travelling Salesperson problems have been studied for decades. Existing formulations assume either a static graph or a temporal graph in which edges are available based on some function of time. In this paper, we introduce a new variant of these problems inspired by applications such as open-pit mining, harvesting and painting, in which some edges become deleted or untraversable depending on the vertices that are visited. We formally define these problems and provide both a theoretical and experimental analysis of them in comparison with the conventional versions. We also propose two solvers, based on an exact backward search and a meta-heuristic solver, and provide an extensive experimental evaluation.

High capacity NP-Complete problems solver based on dual-comb asynchronous optical sampling

Photonic schemes to solve the Nondeterministic Polynomial-Complete (NP–C) problems could offer an interesting alternative to such computationally intensive tasks. Previous studies had shown that they can be solved by measuring the propagation delays through an optical oracle that corresponds to the NP-C problem. However, to tackle a problem with a large instance, the previous schemes face significant challenges to achieve a large measurement range, fast speed and high resolution, simultaneously. In this work, we propose an optical solver based on asynchronous optical sampling driven by a simple fiber dual-comb laser. It is experimentally demonstrated that it can realize picosecond resolution and tens of nanosecond range with a fiber oracle network consisting of 15 couplers and sections of fibers that represents a typical Traveling Salesman problem. With a width of the pulse interferogram of less than 1.5 ps, i.e. an optical path resolution of ∼0.3 mm in fiber, the output from an oracle with 58 likely paths is measured accurately, from which the Hamilton path is easily identified in ∼0.7 ms. It is expected that further optimizing the parameters of the laser as well as the oracle would enable the solving larger scale NP-C problems.

Signed mosaic graphs and alternating mosaic number of knots

Lomonaco and Kauffman introduced knot mosaics in their work on quantum knots. This definition is intended to represent an actual physical quantum system. A knot n-mosaic is an n×n matrix of 11 kinds of specific mosaic tiles representing a knot or a link. In this paper, we consider the alternating mosaic number of an alternating knot K which is defined as the smallest integer n for which K is representable as a reduced alternating knot n-mosaic. We define a signed mosaic graph and a diagonal grid graph and construct Hamiltonian cycles derived from the diagonal grid graphs. Using the cycles, we completely determine the alternating mosaic number of torus knots of type (2,q) for q≥2, which grows in an order of q1/2.

Hamilton cycles in dense regular digraphs and oriented graphs

We prove that for every ε>0 there exists n0=n0(ε) such that every regular oriented graph on n>n0 vertices and degree at least (1/4+ε)n has a Hamilton cycle. This establishes an approximate version of a conjecture of Jackson from 1981. We also establish a result related to a conjecture of Kühn and Osthus about the Hamiltonicity of regular directed graphs with suitable degree and connectivity conditions.

Hamiltonian cycle embedding with fault-tolerant edges and adaptive diagnosis in half hypercube

Hamiltonian cycle embedding with fault-tolerant edges and adaptive diagnosis in half hypercube

Many Hamiltonian subsets in large graphs with given density

AbstractA set of vertices in a graph is a Hamiltonian subset if it induces a subgraph containing a Hamiltonian cycle. Kim, Liu, Sharifzadeh, and Staden proved that for large $d$ , among all graphs with minimum degree $d$ , $K_{d+1}$ minimises the number of Hamiltonian subsets. We prove a near optimal lower bound that takes also the order and the structure of a graph into account. For many natural graph classes, it provides a much better bound than the extremal one ( $\approx 2^{d+1}$ ). Among others, our bound implies that an $n$ -vertex $C_4$ -free graph with minimum degree $d$ contains at least $n2^{d^{2-o(1)}}$ Hamiltonian subsets.

Hybridization of Evolutionary Algorithm and Deep Reinforcement Learning for Multiobjective Orienteering Optimization

Multi-objective orienteering problems (MO-OPs) are classical multi-objective routing problems and have received much attention in recent decades. This study seeks to solve MO-OPs through a problem-decomposition framework, that is, an MO-OP is decomposed into a multi-objective knapsack problem (MOKP) and a traveling salesman problem (TSP). The MOKP and TSP are then solved by a multi-objective evolutionary algorithm (MOEA) and a deep reinforcement learning (DRL) method, respectively. While the MOEA module is for selecting cities, the DRL module is for planning a Hamiltonian path for these cities. An iterative use of these two modules drives the population towards the Pareto front of MO-OPs. The effectiveness of the proposed method is compared against NSGA-II and NSGA-III on various types of MO-OP instances. Experimental results show that our method performs best on almost all the test instances, and has shown strong generalization ability.

Powers of Hamilton cycles in dense graphs perturbed by a random geometric graph

Let G be a graph obtained as the union of some n-vertex graph Hn with minimum degree δ(Hn)≥αn and a d-dimensional random geometric graph Gd(n,r). We investigate under which conditions for r the graph G will a.a.s. contain the kth power of a Hamilton cycle, for any choice of Hn. We provide asymptotically optimal conditions for r for all values of α, d and k. This has applications in the containment of other spanning structures, such as F-factors.

Energy Cost Minimization in Wireless Rechargeable Sensor Networks

Mobile chargers (MCs) are usually dispatched to deliver energy to sensors in wireless rechargeable sensor networks (WRSNs) due to its flexibility and easy maintenance. This paper concerns the fundamental issue of charging path DEsign with the Minimized energy cOst (DEMO), i.e., given a set of rechargeable sensors, we appropriately design the MC’s charging path to minimize the energy cost which is due to the wireless charging and the MC’s movement, such that the different charging demand of each sensor is satisfied. Solving DEMO is NP-hard and involves handling the tradeoff between the charging efficiency and the moving cost. To address DEMO, we first investigate how to identify a single charging position where the MC could stay to charge a set of sensors distributed within a small area with the maximized charging efficiency. Then, based on the result obtained in the case of optimizing a single charging position, we develop a computational geometry-based algorithm to deploy multiple charging positions within the whole network, by considering the fixed and finite charging range of the MC. We prove that the designed algorithm has the approximation ratio of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O\!\left(\ln\!N\right)$</tex-math> </inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$</tex-math> </inline-formula> is the number of sensors. Then we construct the charging path by calculating the shortest Hamiltonian cycle passing through all the deployed charging positions within the network. In addition, we investigate the impact of the network topology as well as the distribution of charging demands among sensors on the MC’s energy cost during a charging tour. Extensive evaluations validate the superiority of our path design in terms of the MC’s energy cost minimization, compared with existing main algorithms.

Research and Design of Calibration Device for Medical Test Lung

In order to solve the problems of quality control and traceability of medical test lung for meeting the calibration conditions of JJF 1234-2018 Calibration Specification for Ventilators, the calibration device and method are researched for compliance and airway resistance of medical test lung in this paper. A calibration device for medical test lung is designed using constant volume active piston technology to simulate human breathing. Through comparison experiment, the deviation between this device and the similar foreign device can be found. The deviation is lower than 0.4% for lung compliance and lower than 0.7% for airway resistance. The calibration of lung compliance and airway resistance can be completed by this device. This device has a clear and complete traceability path to ensure quality control from the source. The calibration of ventilator is improved. This paper provides a reference for related metrology departments and medical institutions to study on quality inspection of respiratory medical instruments.

Hamilton cycles passing through a matching in a bipartite graph with high degree sum

Let G be a balanced bipartite graph of order 2n with partite sets X and Y, and M be a matching of k edges in G. Let σ1,1(G)=min⁡{dG(x)+dG(y):x∈X,y∈Y,xy∉E(G)}. Conditions on σ1,1(G) for the existence of a Hamilton cycle passing through M have been studied by many researchers, but the threshold is not known to be sharp for 2n3≤k≤n−2.In this paper we prove that G has a Hamilton cycle passing through all the edges of M if σ1,1(G)≥2n−k and k≤n−2. Moreover, we show that the lower bound on σ1,1(G) is sharp for every k with 2n3≤k≤n−2.

Disjointness graphs of segments in [formula omitted] are almost all hamiltonian

Let P be a set of n≥1 points in general position in R2. The edge disjointness graphD(P) of P is the graph whose vertices are all the segments with endpoints in P, two of which are adjacent in D(P) if and only if they are disjoint. In this note, we give a full characterization of all those edge disjointness graphs that are hamiltonian. More precisely, we shall show that there are exactly 9 order types of P for which D(P) is not hamiltonian. Additionally, from one of these 9 order types, we derive a counterexample to a criterion for the existence of hamiltonian cycles due to A. D. Plotnikov in 1998.