Hamiltonian Cycle Research Articles

Photon-number moments and cumulants of Gaussian states

We develop closed-form expressions for the moments and cumulants of Gaussian states when measured in the photon-number basis. We express the photon-number moments of a Gaussian state in terms of the loop Hafnian, a function that when applied to a (0,1)-matrix representing the adjacency of a graph, counts the number of its perfect matchings. Similarly, we express the photon-number cumulants in terms of the Montrealer, a newly introduced matrix function that when applied to a (0,1)-matrix counts the number of Hamiltonian cycles of that graph. Based on these graph-theoretic connections, we show that the calculation of photon-number moments and cumulants are #P-hard. Moreover, we provide an exponential time algorithm to calculate Montrealers (and thus cumulants), matching well-known results for Hafnians. We then demonstrate that when a uniformly lossy interferometer is fed in every input with identical single-mode Gaussian states with zero displacement, all the odd-order cumulants but the first one are zero. Finally, we employ the expressions we derive to study the distribution of cumulants up to the fourth order for different input states in a Gaussian boson sampling setup where K identical states are fed into an ℓ-mode interferometer. We analyze the dependence of the cumulants as a function of the type of input state, squeezed, lossy squeezed, squashed, or thermal, and as a function of the number of non-vacuum inputs. We find that thermal states perform much worse than other classical states, such as squashed states, at mimicking the photon-number cumulants of lossy or lossless squeezed states.

One-to-one Disjoint-path Covers of Leaf-sort Graphs

Parallel paths of interconnection networks have attracted much attention in parallel computing systems, they are studied in terms of disjoint paths in an undirected graph [Formula: see text]. A [Formula: see text]-disjoint path cover (k-DPC for short) of a graph [Formula: see text] is a set of [Formula: see text] (internally) disjoint paths that altogether cover every vertice of the graph. A bipartite graph [Formula: see text] is one-to-one bi-k-DPC if there is a k-DPC between any pair of vertices in different partite sets. A bipartite graph is hamiltonian laceable if there exists a hamiltonian path between any pair of vertices in different partite sets. The [Formula: see text]-dimensional leaf-sort graph [Formula: see text] has many good properties, including bipartite, symmetry, vertex-transitive. In this paper, we prove that [Formula: see text] has one-to-one bi-[Formula: see text]-DPC between any two vertices [Formula: see text] and [Formula: see text] in [Formula: see text], where [Formula: see text], [Formula: see text] in different partite sets, and [Formula: see text] is hamiltonian laceable.

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.

A kind of matchings extend to hamiltonian cycles in hypercubes

Ruskey and Savage asked the following question: Does every matching in $Q_{n}$ for $n\geq2$ extend to a Hamiltonian cycle of $Q_{n}$? Kreweras conjectured that every perfect matching of $Q_n$ for $n\geq2$ can be extended to a Hamiltonian cycle of $Q_n$. Fink confirmed the conjecture. An edge in $Q_n$ is an edge of direction $i$ if its endpoints differ in the $i$th position. So all the edges of $Q_n$ can be divided into $n$ directions, i.e., edges of direction $1$, $\ldots$, edges of direction $n$. The set of all edges of direction $i$ of $Q_n$ is denoted by $E_i$. In this paper, we obtain the following result. For $n\geq6$, let $M$ be a matching in $Q_n$ with $|M|<10\times2^{n-5}$. If $M$ contains edges in at most 5 directions, then $M$ can be extended to a Hamiltonian cycle of $Q_n$.

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

Walker-Breaker Games on $G_{n,p}$

The Maker-Breaker connectivity game and Hamilton cycle game belong to the best studied games in positional games theory, including results on biased games, games on random graphs and fast winning strategies. Recently, the Connector-Breaker game variant, in which Connector has to claim edges such that her graph stays connected throughout the game, as well as the Walker-Breaker game variant, in which Walker has to claim her edges according to a walk, have received growing attention.For instance, London and Pluhár studied the threshold bias for the Connector-Breaker connectivity game on a complete graph $K_n$, and showed that there is a big difference between the cases when Maker's bias equals $1$ or $2$. Moreover, a recent result by the first and third author as well as Kirsch shows that the threshold probability $p$ for the $(2:2)$ Connector-Breaker connectivity game on a random graph $G\sim G_{n,p}$ is of order $n^{-2/3+o(1)}$. We extent this result further to Walker-Breaker games and prove that this probability is also enough for Walker to create a Hamilton cycle.

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

AbstractA graph is said to be uniquely hamiltonian if it has a unique hamiltonian cycle. For a natural extension of this concept to infinite graphs, we find all uniquely hamiltonian vertex‐transitive graphs with finitely many ends, and also discuss some examples with infinitely many ends. In particular, we show each nonabelian free group has a Cayley graph of degree that has a unique hamiltonian circle. (A weaker statement had been conjectured by Georgakopoulos.) Furthermore, we prove that these Cayley graphs of are outerplanar.

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.