Hamiltonian Cycle Research Articles

The Chvátal–Gomory procedure for integer SDPs with applications in combinatorial optimization

AbstractIn this paper we study the well-known Chvátal–Gomory (CG) procedure for the class of integer semidefinite programs (ISDPs). We prove several results regarding the hierarchy of relaxations obtained by iterating this procedure. We also study different formulations of the elementary closure of spectrahedra. A polyhedral description of the elementary closure for a specific type of spectrahedra is derived by exploiting total dual integrality for SDPs. Moreover, we show how to exploit (strengthened) CG cuts in a branch-and-cut framework for ISDPs. Different from existing algorithms in the literature, the separation routine in our approach exploits both the semidefinite and the integrality constraints. We provide separation routines for several common classes of binary SDPs resulting from combinatorial optimization problems. In the second part of the paper we present a comprehensive application of our approach to the quadratic traveling salesman problem (QTSP). Based on the algebraic connectivity of the directed Hamiltonian cycle, two ISDPs that model the QTSP are introduced. We show that the CG cuts resulting from these formulations contain several well-known families of cutting planes. Numerical results illustrate the practical strength of the CG cuts in our branch-and-cut algorithm, which outperforms alternative ISDP solvers and is able to solve large QTSP instances to optimality.

Finding triangle‐free 2‐factors in general graphs

AbstractA 2‐factor in a graph is a subset of edges such that every node of is incident with exactly two edges of . Many results are known concerning 2‐factors including a polynomial‐time algorithm for finding 2‐factors and a characterization of those graphs that have a 2‐factor. The problem of finding a 2‐factor in a graph is a relaxation of the NP‐hard problem of finding a Hamilton cycle. A stronger relaxation is the problem of finding a triangle‐free 2‐factor, that is, a 2‐factor whose edges induce no cycle of length 3. In this paper, we present a polynomial‐time algorithm for the problem of finding a triangle‐free 2‐factor as well as a characterization of the graphs that have such a 2‐factor and related min–max and augmenting path theorems.

Embedding Hamiltonian Paths with Prescribed Linear Forests into k-ary n-Cube Networks

Graph embedding is a fundamental problem in computer science. Let [Formula: see text] (resp., [Formula: see text]) be the interconnection network for a parallel computer system [Formula: see text] (resp., [Formula: see text]). If [Formula: see text] could be embedded into [Formula: see text], then [Formula: see text] can simulate [Formula: see text]’s behavior. The [Formula: see text]-ary [Formula: see text]-cube [Formula: see text] is a node-symmetric and link-symmetric recursive interconnection network for parallel computer systems. Let [Formula: see text] be a prescribed linear forest of [Formula: see text], and let [Formula: see text] and [Formula: see text] be any two distinct nodes in [Formula: see text] such that [Formula: see text] has no path with [Formula: see text] or [Formula: see text] as internal nodes, or both as end-nodes. This paper shows that there is a Hamiltonian path passing through [Formula: see text] between [Formula: see text] and [Formula: see text] in [Formula: see text] with [Formula: see text] and odd [Formula: see text] even if the number of links in [Formula: see text] is up to [Formula: see text].

The Hamiltonian p-median problem: Polyhedral results and branch-and-cut algorithms

In this paper we study the Hamiltonian p-median problem, in which we are given an edge-weighted graph and we are asked to determine p vertex-disjoint cycles spanning all vertices of the graph and having minimum total weight. We introduce two new families of valid inequalities for a formulation of the problem in the space of edge variables. Each one of the families forbids solutions to the 2-factor relaxation of the problem that have less than p cycles. The inequalities in one of the families are associated with large cycles of the underlying graph and generalize known inequalities associated with Hamiltonian cycles. The other family involves inequalities for the case with p=n/3, associated with edge cuts and multi-cuts whose shores have specific cardinalities. We identify inequalities from both families that define facets of the polytope associated with the problem. We design branch-and-cut algorithms based on these families of inequalities and on inequalities associated with 2-opt moves removing sub-optimal solutions. Computational experiments on benchmark instances show that the proposed algorithms exhibit a comparable performance with respect to existing exact methods from the literature. Moreover the algorithms solve to optimality new instances with up to 400 vertices.

On Hamiltonian Cycles in a 2-Strong Digraphs with Large Degrees and Cycles

On Hamiltonian Cycles in a 2-Strong Digraphs with Large Degrees and Cycles

A Study on Emergence of Clutch Graphs from Cycle Graphs: A Comprehensive Analysis

The clutch is one of the substantial devices for constructing vehicles in automobile engineering. In terms of network toughness against failures or disruptions, the clutch-based graph can be used to model insufficiency of primary pathways and the availability of alternative routes in communication networks. In this paper, the identification of the clutch graph Cl3n(G) from the cycle graph Cn(G) has been proposed. A clutch graph Cl3n(G) generating from a cycle graph with 3n vertices and 4n edges (n ≥ 4 and even). The notions of degree, girth, and chromatic number of the clutch graph have been discussed. Further, the existence of the bipartite and Hamiltonian graphs on the clutch graph has been examined.

Some conditions for hamiltonian cycles in 1-tough (K2 ∪ kK1)-free graphs

Let k≥2 be an integer. We say that a graph G is (K2∪kK1)-free if it does not contain K2∪kK1 as an induced subgraph. Recently, Shi and Shan conjectured that every 1-tough and 2k-connected (K2∪kK1)-free graph is hamiltonian. In this paper, we solve this conjecture by proving that every 1-tough and k-connected (K2∪kK1)-free graph with minimum degree at least 3(k−1)2 is hamiltonian or the Petersen graph.

Proving the existence of Euclidean knight’s tours on n × n × ⋯ × n chessboards for n < 4

The Knight's Tour problem consists of finding a Hamiltonian path for the knight on a given set of points so that the knight can visit exactly once every vertex of the mentioned set. In the present, we provide a 5-dimensional alternative to the well-known statement that it is not ever possible for a knight to visit once every vertex of $C(3,k):=\{\underbrace{\{0,1,2\} \times \{0,1,2\}\times \cdots \times \{0,1,2\}}_\textrm{\textit{k}-times}\}$ by performing a sequence of $3^k-1$ jumps of standard length, since the most accurate answer to the original question actually depends on which mathematical assumptions we are making at the beginning of the game when we decide to extend a planar chess piece to the third dimension and above. Our counterintuitive outcome follows from the observation that we can alternatively define a 2D knight as a piece that moves from one square to another on the chessboard by covering a fixed Euclidean distance of $\sqrt{5}$ so that also the statement of Theorem 3 in [Erde, J., Golénia, B., & Golénia, S. (2012), The closed knight tour problem in higher dimensions, The Electronic Journal of Combinatorics, 19(4), #P9] does not hold anymore for such a Euclidean knight, as long as a 2 × 2 × ⋯ × 2 chessboard with at least 2<sup>7</sup> cells is given. Moreover, we construct a classical closed knight's tour on $C(3,4)-\{(1,1,1,1)\}$ whose arrival is at a distance of 2 from $(1,1,1,1)$, and then we show a closed Euclidean knight's tour on $\{\{0,1\}\times\{0,1\}\times\{0,1\}\times\{0,1\}\times\{0,1\}\times\{0,1\}\times\{0,1\}\}\subseteq \mathbb{Z}^7$.

Using path coloring of graphs for communication in social networks

An L (p1, p2, p3, … , pm)- labeling of a graph G, has the vertices of G assigned with non-negative integers, such that the vertices at distance j should have at least pj as their label difference. If m = 3 and p1 = 3, p2 = 2, p3 = 1, it is called an L (3, 2, 1)-labeling which is widely studied in the literature. In this paper, we define an L (3, 2, 1)-path coloring of G as a labeling g : V (G) → Z+ such that between every pair of vertices there exists at least one path P where in the labeling restricted to this path is an L (3, 2, 1)-labeling. Among the labels assigned to any vertex of G under g, the maximum label is called the span of g. The L (3, 2, 1)-connection number of a graph G, denoted by k3c (G) is defined as the minimum value of span of g taken over all such labelings g. We call graphs with the special property that k3c (G) = |V (G) | as L (3, 2, 1)-path graceful. In this paper, we obtain k3c (G) of graphs that possess a Hamiltonian path and carry forward the discussion to certain classes of graphs which do not possess a Hamiltonian path, which is novel to this paper. Although different kinds of labeling are studied in the literature with different mathematical constraints imposed, the idea of showing the existence of a graph with a given number as its minimum labeling number has rarely been addressed. We show that given any positive integer, there always exists an L (3, 2, 1)-path graceful graph with the given integer as its k3c (G), thus addressing the inverse question. Finally exploiting the fact that there is no gap on the k3c (G) number line, we give an application of path colorings for secure communication on social networking sites. Efforts to deploy graph coloring in task scheduling, interference-free transmission, etc have been dealt by earlier researchers. In this paper, we deploy the L (3, 2, 1)-path coloring technique defined by us for secure communication in social networks, which has not been dealt with so far.

An extension of the Christofides heuristic for a single-depot multiple Hamiltonian path problem

An extension of the Christofides heuristic for a single-depot multiple Hamiltonian path problem

CONDITIONS FOR GRAPHS ON <i>n</i> VERTICES WITH THE SUM OF DEGREES OF ANY TWO NONADJACENT VERTICES EQUAL TO<i> n-2</i> TO BE A HAMILTONIAN GRAPH

Let G be an undirected simple graph on \(n \geq 3\) vertices with the degree sum of any two nonadjacent vertices in G equal to \(n - 2\). We determine the condition for G to be a Hamiltonian graph.

Few hamiltonian cycles in graphs with one or two vertex degrees

Inspired by Sheehan’s conjecture that no 4 4 -regular graph contains exactly one hamiltonian cycle, we prove results on hamiltonian cycles in regular graphs and nearly regular graphs. We fully disprove a conjecture of Haythorpe on the minimum number of hamiltonian cycles in regular hamiltonian graphs, thereby extending a result of Zamfirescu, as well as correct and complement Haythorpe’s computational enumerative results from [Exp. Math. 27 (2018), no. 4, 426–430]. Thereafter, we use the Lovász Local Lemma to extend Thomassen’s independent dominating set method. This extension allows us to find a second hamiltonian cycle that inherits linearly many edges from the first hamiltonian cycle. Regarding the limitations of this method, we answer a question of Haxell, Seamone, and Verstraete, and settle the first open case of a problem of Thomassen by showing that for k ∈ { 5 , 6 } k \in \{5, 6\} there exist infinitely many k k -regular hamiltonian graphs having no independent dominating set with respect to a prescribed hamiltonian cycle. Motivated by an observation of Aldred and Thomassen, we prove that for every κ ∈ { 2 , 3 } \kappa \in \{ 2, 3 \} and any positive integer k k , there are infinitely many non-regular graphs of connectivity κ \kappa containing exactly one hamiltonian cycle and in which every vertex has degree 3 3 or 2 k 2k .

Compatible Spanning Circuits and Forbidden Induced Subgraphs

A compatible spanning circuit in an edge-colored graph G (not necessarily properly) is defined as a closed trail containing all vertices of G in which any two consecutively traversed edges have distinct colors. The existence of extremal compatible spanning circuits (i.e., compatible Hamilton cycles and compatible Euler tours) has been studied extensively. Recently, sufficient conditions for the existence of compatible spanning circuits visiting each vertex at least a specified number of times in specific edge-colored graphs satisfying certain degree conditions have been established. In this paper, we continue the research on sufficient conditions for the existence of such compatible s-panning circuits. We consider edge-colored graphs containing no certain forbidden induced subgraphs. As applications, we also consider the existence of such compatible spanning circuits in edge-colored graphs G with κ(G) ≥ α(G), κ(G) ≥ α(G) − 1 and κ (G) ≥ α(G), respectively. In this context, κ(G), α(G) and κ (G) denote the connectivity, the independence number and the edge connectivity of a graph G, respectively.

Hybrid ITÖ Algorithm for Maximum Scatter Colored Traveling Salesman Problem

Abstract This research presents a new problem maximum scatter colored traveling salesman problem (MSCTSP), the objective of MSCTSP is to find Hamiltonian cycles with the minimal edge as max as possible, it is used to simulate the real-world applications of network and transport. Since MSCTSP has been proved to be a NP-hard problem, population-based algorithms can be used for solving it. However, the performances are not satisfactory. Thus, it is necessary to develop novel algorithms to obtain high quality feasible solution. Based on the above reason, the paper proposes a novel hybrid ITÖ (HITÖ) algorithm, which integrates the two new strategies: crossover operator and mutation strategy, into the standard ITÖ. In the iteration course of HITÖ, the dual-chromosome coding is used to code a feasible solution of MSCTSP, and the stochastic drift and volatility operators are used to explore and exploit new unknown region. During the process, drift operator is performed by crossover operator, volatility operator is carried out by mutation strategy, and they are both affected by activity intensity of particles which functionally depends on the radius and temperature. Experiments display HITÖ shows an improvement over comparative algorithms on solution quality.

Power of $k$ Choices in the Semi-Random Graph Process

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible.
 In this paper, we introduce a natural generalization of this game in which $k$ random vertices $u_1, \ldots, u_k$ are presented to the player in each round. She needs to select one of the presented vertices and connect to any vertex she wants. We focus on the following three monotone properties: minimum degree at least $\ell$, the existence of a perfect matching, and the existence of a Hamiltonian cycle.

A hierarchical structure of the quad-cube and the associated diameter, domination, and Hamiltonicity

The quad-cube is a special case of the metacube that itself is derivable from the hypercube. It is amenable to an application as a network topology, especially when the node size exceeds several million. This paper presents a sequence of graphs, leading up to the quad-cube, where the graphs in the sequence are important in their own right. For example, they exhibit hypercube-like low diameters and efficient domination parameters, and most of them admit a Hamiltonian cycle. The hierarchy is likely to be useful in the future.

On degree conditions of semi-balanced k-partite Hamiltonian graphs

For two integers [Formula: see text] and [Formula: see text] with [Formula: see text], a [Formula: see text]-partite graph is said to be a semi-balanced[Formula: see text]-partite graph if each partite set has either [Formula: see text] or [Formula: see text] vertices. We give some degree conditions for semi-balanced [Formula: see text]-partite graphs to be Hamiltonian and almost pancyclic, respectively. Furthermore, using these results, we obtain the degree conditions for a balanced [Formula: see text]-partite graph to be vertex bipancyclic if [Formula: see text] is even, and vertex [Formula: see text]-pancyclic if [Formula: see text] is odd.

Beyond Quasi-Adjoint Graphs: On Polynomial-Time Solvable Cases of the Hamiltonian Cycle and Path Problems

The Hamiltonian cycle and path problems are fundamental in graph theory and useful in modelling real-life problems. Research in this area is directed toward designing better and better algorithms for general problems, but also toward defining new special cases for which exact polynomial-time algorithms exist. In the paper, such new classes of digraphs are proposed. The classes include, among others, quasi-adjoint graphs, which are a superclass of adjoints, directed line graphs, and graphs modelling a DNA sequencing problem.

Enumeration of dissociation sets in grid graphs

<abstract><p>A dissociation set of a graph $ G $ refers to a set of vertices inducing a subgraph with maximum degree at most 1 and serves as a generalization of two fundamental concepts in graph theory: Independent sets and induced matchings. The enumeration of specific substructures in grid graphs has been a captivating area of research in graph theory. Over the past few decades, the enumeration problems related to various structures in grid graphs such as Hamiltonian cycles, Hamiltonian paths, independent sets, maximal independent sets, and dominating sets have been deeply studied. In this paper, we enumerated dissociation sets in grid graphs using the state matrix recursion algorithm.</p></abstract>

Fault-tolerant Hamiltonian cycle strategy for fast node fault diagnosis based on PMC in data center networks.

System-level fault diagnosis model, namely, the PMC model, detects fault nodes only through the mutual testing of nodes in the system without physical equipment. In order to achieve server nodes fault diagnosis in large-scale data center networks (DCNs), the traditional algorithm based on the PMC model cannot meet the characteristics of high diagnosability, high accuracy and high efficiency due to its inability to ensure that the test nodes are fault-free. This paper first proposed a fault-tolerant Hamiltonian cycle fault diagnosis (FHFD) algorithm, which tests nodes in the order of the Hamiltonian cycle to ensure that the test nodes are faultless. In order to improve testing efficiency, a hierarchical diagnosis mechanism was further proposed, which recursively divides high scale structures into a large number of low scale structures based on the recursive structure characteristics of DCNs. Additionally, we proved that $ 2(n-2){n^{k-1}} $ and $ (n-2){t_{n, k}}/{t_{n, 1}} $ faulty nodes could be detected for $ BCub{e_{n, k}} $ and $ DCel{l_{n, k}} $ within a limited time for the proposed diagnosis strategy. Simulation experiments have also shown that our proposed strategy has improved the diagnosability and test efficiency dramatically.