Hamiltonian Cycle Research Articles

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.

Hamiltonicity of Schrijver graphs and stable Kneser graphs

For integers k≥1 and n≥2k+1, the Schrijver graph S(n,k) has as vertices all k-element subsets of [n]≔{1,2,…,n} that contain no two cyclically adjacent elements, and an edge between any two disjoint sets. More generally, for integers k≥1, s≥2, and n≥sk+1, the s-stable Kneser graph S(n,k,s) has as vertices all k-element subsets of [n] in which any two elements are in cyclical distance at least s. We prove that all the graphs S(n,k,s), in particular Schrijver graphs S(n,k)=S(n,k,2), admit a Hamilton cycle that can be computed in time O(n) per generated vertex.

Solving QUBO problems with cP systems

P systems with compound terms (cP systems) have been proposed by Radu Nicolescu in 2018. These expressive cP systems have been used to solve well-known NP-complete problems efficiently, such as the Hamiltonian path, traveling salesman, 3-coloring, and software verification problems. In this paper, we use cP systems to provide an efficient parallel solution to the integer-valued quadratic unconstrained Boolean optimization (QUBO) problem.

Spanning cycles in random directed graphs

AbstractWe show that, in almost every ‐vertex random directed graph process, a copy of every possible ‐vertex oriented cycle will appear strictly before a directed Hamilton cycle does, except of course for the directed cycle itself. Furthermore, given an arbitrary ‐vertex oriented cycle, we determine the sharp threshold for its appearance in the binomial random directed graph. These results confirm, in a strong form, a conjecture of Ferber and Long.

Order Structured Graphs of Cyclic Groups and their Classification

Let $\Gamma^{o}(G)$ with $G\cong C_{p},$ a cyclic group of order $p,$ be an order structured graph. The group $C_{p}$ will be assumed as the vertex set of the graph $\Gamma^{o}(G)$ and an edge between vertices will be built on the basis of a defined relation via order structure. Certain graphical parameters such as independence ratio, clique number, domination number, and separability are discussed. Some characterizations are proposed and proved by incorporating the defined relation. It is further proved that $\Gamma^{o}(C_{p})$ can never be a hamiltonian graph. Lastly, It is shown that $C(\Gamma^{o}(C_{p}))$ is isomorphic to $\Gamma^{o}(C_{p}).

Modified Detour Index of Hamiltonian Connected (Laceable) Graphs

Objectives: To explore the bounds for the modified detour index of certain Hamiltonian connected and laceable graphs. Methods: The Wiener index , detour index and the modified detour index are used. Findings: Here we introduce the modified detour index and its least upper bounds for Hamiltonian connected and laceable graphs, by formulating the constraints. Novelty: Based on the modified detour index, the bounds for some special graphs such as: Hamiltonian connected graphs of two families of convex polytopes ( and ) and Hamiltonian laceable graphs of spider graph ( ) and image graph of prism graph ( ) are encountered here. Keywords: Hamiltonian graph, Hamiltonian connected, Hamiltonian laceable, Wiener index, detour index

Rainbow Hamiltonicity in uniformly coloured perturbed digraphs

AbstractWe investigate the existence of a rainbow Hamilton cycle in a uniformly edge-coloured randomly perturbed digraph. We show that for every $\delta \in (0,1)$ there exists $C = C(\delta ) \gt 0$ such that the following holds. Let $D_0$ be an $n$ -vertex digraph with minimum semidegree at least $\delta n$ and suppose that each edge of the union of $D_0$ with a copy of the random digraph $\mathbf{D}(n,C/n)$ on the same vertex set gets a colour in $[n]$ independently and uniformly at random. Then, with high probability, $D_0 \cup \mathbf{D}(n,C/n)$ has a rainbow directed Hamilton cycle.This improves a result of Aigner-Horev and Hefetz ((2021) SIAM J. Discrete Math.35(3) 1569–1577), who proved the same in the undirected setting when the edges are coloured uniformly in a set of $(1 + \varepsilon )n$ colours.

An effective graph-analysis method to schedule a continuous galvanizing line with campaigning boundary constraints

This work deals with the scheduling optimization of a Continuous Galvanizing Line (CGL) in the steel industry. We introduce a real-case extension of the CGL scheduling problem that is concerned with the linking of campaigns. In order to link the different campaigns planned at the CGL, the scheduler may fix the start coil of the sequence, the end coil, or both, introducing new boundary constraints (BC) to the sequencing problem. This shows to have a significant impact in the performance of the current Ant Colony Optimization (ACO) algorithms in use, failing to find feasible solutions with reliability. Our research aims at improving this reliability. Viewing sequences as paths in a directed weighed graph, in which coils are the nodes, and coil changes the edges, the BC problem is to find a minimum cost Hamiltonian path that respects the required start and end nodes. In this paper we study the negative impact brought by the BC in 30 challenging instances from the CGL, we discuss the reasons behind, and we propose a new algorithm able to robustly assure feasibility in all of them. We introduce a brand-new graph analysis method devised as an effective surrogate check for feasibility, which runs embedded in the ACO sequence construction heuristic. In the experimental analysis we see that it clearly boosts performance, increasing reliability by 20%, up to 99.67%. By teaming up with the Interval Reconstruction local search, the reliability and quality of the solutions shows to improve further. This graph analysis method we propose, though illustrated within ACO as a use case, is applicable to any constructive metaheuristic.

Graphs with at most two moplexes

AbstractA moplex is a natural graph structure that arises when lifting Dirac's classical theorem from chordal graphs to general graphs. The notion is known to be closely related to lexicographic searches in graphs as well as to asteroidal triples, and has been applied in several algorithms related to graph classes, such as interval graphs, claw‐free, and diamond‐free graphs. However, while every noncomplete graph has at least two moplexes, little is known about structural properties of graphs with a bounded number of moplexes. The study of these graphs is, in part, motivated by the parallel between moplexes in general graphs and simplicial modules in chordal graphs: unlike in the moplex setting, properties of chordal graphs with a bounded number of simplicial modules are well understood. For instance, chordal graphs having at most two simplicial modules are intervals. In this work, we initiate an investigation of ‐moplex graphs, which are defined as graphs containing at most moplexes. Of particular interest is the smallest nontrivial case , which forms a counterpart to the class of interval graphs. As our main structural result, we show that, when restricted to connected graphs, the class of 2‐moplex graphs is sandwiched between the classes of proper interval graphs and cocomparability graphs; moreover, both inclusions are tight for hereditary classes. From a complexity‐theoretic viewpoint, this leads to the natural question of whether the presence of at most two moplexes guarantees a sufficient amount of structure to efficiently solve problems that are known to be intractable on cocomparability graphs, but not on proper interval graphs. We develop new reductions that answer this question negatively for two prominent problems fitting this profile, namely, Graph Isomorphism and Max‐Cut. On the other hand, we prove that every connected 2‐moplex graph contains a Hamiltonian path, generalising the same property of connected proper interval graphs. Furthermore, for graphs with a higher number of moplexes, we lift the previously known result that graphs without asteroidal triples have at most two moplexes to the more general setting of larger asteroidal sets.

On Hamilton Cycles in Graphs Defined by Intersecting Set Systems

On Hamilton Cycles in Graphs Defined by Intersecting Set Systems

On approximating a new generalization of traveling salesman problem

The current best-known performance guarantees for the extensively studied Traveling Salesman Problem (TSP) of determinate approximation algorithms is 32, achieved by Christofides' algorithm 47 years ago. This paper investigates a new generalization problem of the TSP, termed the Minimum-Cost Bounded Degree Connected Subgraph (MBDCS) problem. In the MBDCS problem, the goal is to identify a minimum-cost connected subgraph containing n=|V| edges from an input graph G=(V,E) with degree upper bounds for particular vertices. We show that for certain special cases of MBDCS, the aim is equivalent to finding a minimum-cost Hamiltonian cycle for the input graph, same as the TSP. To appropriately solve MBDCS, we initially present an integer programming formulation for the problem. Subsequently, we propose an algorithm to approximate the optimal solution by applying the iterative rounding technique to solution of the integer programming relaxation. We demonstrate that the returned subgraph of our proposed algorithm is one of the best guarantees for the MBDCS problem in polynomial time, assuming P≠NP. This study views the optimization of TSP as finding a minimum-cost connected subgraph containing n edges with degree upper bounds for certain vertices, and it may provide new insights into optimizing the TSP in future research.

Sustainable Time-Dependent Cheapest Path Problem with Integrated Collaborative Stakeholders’ Perspectives

The Sustainable Time-Dependent Cheapest Path Problem (STDCPP) entails locating a Hamiltonian path that covers all of the graph’s vertices at the lowest possible total sustainability cost. The issue is inspired by actual city logistics, where it is important to consider the opinions of diverse stakeholders in the light of sustainable urban mobility plans and service viability. To address this issue, this paper suggests a twofold contribution. First, we describe the Sustainable Time-Dependent Cheapest Path Problem and define the complex cost function, which, based on the multi-criteria decision-making approach, integrates the views of different stakeholders and sustainability elements into the route cost calculation. Second, we show that the modified problem satisfies the FIFO (First-In First-Out) property and demonstrate the applicability of the suggested approach on a real-life scenario where route sustainability is extracted from the traffic sign information system available in Flanders, Belgium.

Interplays between variations of arbitrarily partitionable graphs under minimality constraints

An arbitrarily partitionable (AP) graph is a graph that can be partitioned into arbitrarily many connected graphs with arbitrary orders. Since independent seminal works by Barth, Baudon, and Puech, and Horňák and Woźniak, AP graphs have been receiving increasing attention in the literature, dedicated to understanding several of their aspects, including structural aspects, algorithmic aspects, and their connections with Hamiltonian graphs. Other aspects of interest cover variants of AP graphs, such as AP graphs that can be partitioned in an online way (OLAP graphs), AP graphs that can be partitioned in a recursive way (RAP graphs), and AP graphs that are edge-minimal (minAP graphs).In the current work, we initiate the study of the latter notion of minimality for OLAP and RAP graphs. That is, we wonder about OLAP and RAP graphs that are not spanned by any strictly smaller OLAP or RAP graph, respectively, leading to the notions of minOLAP and minRAP graphs. We prove that such non-trivial graphs exist, and explore connections between minAPness, minOLAPness, and minRAPness. In particular, we prove that some fundamental connections between APness, OLAPness, and RAPness do not generalise to their minimal counterparts. We also investigate small minOLAP and minRAP graphs, as well as sufficient conditions guaranteeing an OLAP or RAP graph is not minimal, thereby generalising known results on AP graphs. This work also includes many open questions and problems for further work on the topic, that are disseminated throughout.

A three-phase algorithm for the pollution traveling Salesman problem

This paper studies a variant of the Pollution Traveling Salesman Problem (PTSP) focused on fuel consumption and pollution emissions (PTSPC). The PTSPC generalizes the well-known Traveling Salesman Problem (TSP), classified as NP-Hard. In the PTSPC, a vehicle must deliver a load to each customer through a Hamiltonian cycle, minimizing an objective function that considers the speed of each edge, the mass of the truck, the mass of the load pending delivery, and the distance traveled. We have proposed a three-phase algorithm for the PTSPC. The first phase solves the Traveling Salesman Problem (TSP) exactly with a time limit and heuristically using a Nearest Neighborhood Search approach. This phase considers the constraints associated with the PTSPC by using commercial software. In the second phase, both the obtained solutions and their inverse sequences from the initial phase undergo enhancement utilizing metaheuristic algorithms tailored for the PTSPC. These algorithms include Variable Neighborhood Search (VNS), Tabu Search (TS), and Simulated Annealing (SA). Subsequently, for the third phase, the best solution identified in the second phase—determined by having the minimum value by the PTSPC objective function—is subjected to resolution by a mathematical model designed for the PTSPC, considering the heuristic emphasis of commercial software. The efficiency of the former algorithm has been validated through experimentation involving the adaptation of instances from the Pollution Routing Problem (PRP) to the PTSPC. This approach demonstrates the capacity to yield high-quality solutions within acceptable computing times.

Hamilton Cycles in the Line Graph of a Random Hypergraph

Hamilton Cycles in the Line Graph of a Random Hypergraph

Enabling high fault-tolerant embedding capability of alternating group graphs

The Hamiltonian path/cycle serves as a robust tool for transmitting messages within parallel and distributed systems. However, the prevalent device-intensive nature of these systems often leads to the occurrence of faults. Tackling the critical challenge of tolerating numerous faults when constructing Hamiltonian paths and cycles in these systems is of utmost significance. The alternating group graph AGn is a suitable interconnection network for building parallel and distributed systems due to its outstanding topological properties. Many studies have been dedicated to the fault-tolerant embedding of Hamiltonian paths and cycles in AGn, but they all fail to reach the desired level of fault-tolerance. In this paper, we aim to enhance the edge fault-tolerant embedding capability of AGn by leveraging a newly emerged fault model, called the Partitioned Edge Fault (PEF) model. We prove the existence of Hamiltonian paths/cycles tolerating large-scale edge faults in AGn under the PEF model. Additionally, we propose a fault-tolerant Hamiltonian path embedding algorithm for AGn under the PEF model and validate its effectiveness through experiments. To gauge the significance of each missed edge, we conduct experimental calculations for the average path length of every edge along the Hamiltonian path produced by the proposed embedding algorithm. Furthermore, the fault-tolerance comparison and experimental analysis reveal that our results improve the fault-tolerant embedding capability of AGn from a linear level to an exponential one. To our knowledge, this is the first time that a Hamiltonian path/cycle embedding approach is proposed and actually implemented for AGn with large-scale edge faults.

The square of a Hamilton cycle in randomly perturbed graphs

AbstractWe investigate the appearance of the square of a Hamilton cycle in the model of randomly perturbed graphs, which is, for a given , the union of any ‐vertex graph with minimum degree and the binomial random graph . This is known when and we determine the exact perturbed threshold probability in all the remaining cases, that is, for each . We demonstrate that, as ranges over the interval , the threshold performs a countably infinite number of ‘jumps’. Our result has implications on the perturbed threshold for two‐universality, where we also fully address all open cases.

Transference for loose Hamilton cycles in random 3‐uniform hypergraphs

AbstractA loose Hamilton cycle in a hypergraph is a cyclic sequence of edges covering all vertices in which only every two consecutive edges intersect and do so in exactly one vertex. With Dirac's theorem in mind, it is natural to ask what minimum ‐degree condition guarantees the existence of a loose Hamilton cycle in a ‐uniform hypergraph. For and each , the necessary and sufficient such condition is known precisely. We show that these results adhere to a ‘transference principle’ to their sparse random analogues. The proof combines several ideas from the graph setting and relies on the absorbing method. In particular, we employ a novel approach of Kwan and Ferber for finding absorbers in subgraphs of sparse hypergraphs via a contraction procedure. In the case of , our findings are asymptotically optimal.

A bound on relative lengths of triangle-free graphs

For a 2-connected graph G, the relative length of G, denoted by diff(G), is the difference between the orders of a longest path and a longest cycle in G. This parameter is used as a measure to estimate how close a given graph is to a hamiltonian graph. Let σk(G) be the least value of the sums of degrees of vertices in independent sets of cardinality k. In 2008, Paulusma and Yoshimoto proved that a 2-connected triangle-free graph G of order n with σ4(G)≥n+2 satisfies diff(G)≤1 unless G is isomorphic to one exceptional graph G0. In this paper, we extend their result and prove that for an integer s with 23(n+4)<s≤n+2, a 2-connected triangle-free graph of order n with σ4(G)≥s satisfies diff(G)≤n+3−s unless s=σ4(G)=n+2 and G=G0.