Hamiltonian Cycle Research Articles

Find subtrees of specified weight and cycles of specified length in linear time

AbstractWe apply the Euler tour technique to find subtrees of specified weight as follows. Let such that and , where . Let be a tree of vertices and let be vertex weights such that and for all . We prove that a subtree of of weight exists and can be found in linear time. We apply it to show, among others, the following: Every planar hamiltonian graph with minimum degree has a cycle of length for every with . Every 3‐connected planar hamiltonian graph with and even has a cycle of length or .Each of these cycles can be found in linear time if a Hamilton cycle of the graph is given. This study was partially motivated by conjectures of Bondy and Malkevitch on cycle spectra of 4‐connected planar graphs.

On Hamilton-Connectivity and Detour Index of Certain Families of Convex Polytopes

A convex polytope is the convex hull of a finite set of points in the Euclidean space ℝ n . By preserving the adjacency-incidence relation between vertices of a polytope, its structural graph is constructed. A graph is called Hamilton-connected if there exists at least one Hamiltonian path between any of its two vertices. The detour index is defined to be the sum of the lengths of longest distances, i.e., detours between vertices in a graph. Hamiltonian and Hamilton-connected graphs have diverse applications in computer science and electrical engineering, whereas the detour index has important applications in chemistry. Checking whether a graph is Hamilton-connected and computing the detour index of an arbitrary graph are both NP-complete problems. In this paper, we study these problems simultaneously for certain families of convex polytopes. We construct two infinite families of Hamilton-connected convex polytopes. Hamilton-connectivity is shown by constructing Hamiltonian paths between any pair of vertices. We then use the Hamilton-connectivity to compute the detour index of these families. A family of non-Hamilton-connected convex polytopes has also been constructed to show that not all convex polytope families are Hamilton-connected.

Rainbow Pancyclicity in Graph Systems

Let $G_1,\ldots,G_n$ be graphs on the same vertex set of size $n$, each graph with minimum degree $\delta(G_i)\ge n/2$. A recent conjecture of Aharoni asserts that there exists a rainbow Hamiltonian cycle i.e. a cycle with edge set $\{e_1,\ldots,e_n\}$ such that $e_i\in E(G_i)$ for $1\leq i \leq n$. This can be viewed as a rainbow version of the well-known Dirac theorem. In this paper, we prove this conjecture asymptotically by showing that for every $\varepsilon>0$, there exists an integer $N>0$, such that when $n>N$ for any graphs $G_1,\ldots,G_n$ on the same vertex set of size $n$ with $\delta(G_i)\ge (\frac{1}{2}+\varepsilon)n$, there exists a rainbow Hamiltonian cycle. Our main tool is the absorption technique. Additionally, we prove that with $\delta(G_i)\geq \frac{n+1}{2}$ for each $i$, one can find rainbow cycles of length $3,\ldots,n-1$.

On the intersection graph of the disks with diameters the sides of a convex [formula omitted]-gon

Given a convex n-gon, we can draw n disks (called side disks) where each disk has a different side of the polygon as diameter and the midpoint of the side as its center. The intersection graph of such disks is the undirected graph with vertices the n disks and two disks are adjacent if and only if they have a point in common. Such a graph was introduced by Huemer and Pérez-Lantero (Discrete Mathematics, 2020), proved to be planar and Hamiltonian. In this paper, we study further combinatorial properties of this graph. We prove that the treewidth is at most 3, by showing an O(n)-time algorithm that builds a tree decomposition of width at most 3, given the polygon as input. This implies that we can construct the intersection graph of the side disks in O(n) time. We further study the independence number, which is the maximum number of pairwise disjoint disks. The planarity condition implies that for every convex n-gon we can select at least ⌈n/4⌉ pairwise disjoint disks, and we prove that for every n≥3 there exist convex n-gons in which we cannot select more than this number. Finally, we show that our class of graphs includes all outerplanar Hamiltonian graphs except the cycle of length four, and that it is a proper subclass of the planar Hamiltonian graphs.

Characterizing forbidden subgraphs that imply pancyclicity in 4-connected, claw-free graphs

In 1984, Matthews and Sumner conjectured that every 4-connected, claw-free graph contains a Hamiltonian cycle. This still unresolved conjecture has been the motivation for research into the existence of other cycle structures. In this paper, we consider the stronger property of pancyclicity for 4-connected graphs. In particular, we show that every 4-connected, {K1,3,N(i,j,k)}-free graph, where i,j,k≥1 and i+j+k=6, is pancyclic. This, together with results by Ferrara, Morris, Wenger, and Ferrara et al. completes a characterization of the graphs Y such that every {K1,3,Y}-free graph is pancyclic. In addition, this represents the best known progress towards answering a question of Gould concerning a characterization of the pairs of forbidden subgraphs that imply pancyclicity in 4-connected graphs.

4-connected polyhedra have at least a linear number of hamiltonian cycles

Although polyhedra can have much fewer edges than triangulations, many results about hamiltonicity proven for triangulations also hold for polyhedra. The most famous of these results is surely Whitney’s result from 1931 that 4-connected triangulations are hamiltonian, which was 25 years later generalised to 4-connected polyhedra by Tutte. Nevertheless the only known bounds for the number of hamiltonian cycles in 4-connected polyhedra are constant, though for triangulations a lower bound of |V|∕(log2|V|) was already proven in 1979 and improved to a linear bound in 2018. In this article we prove linear lower bounds for 4-connected polyhedra and polyhedra with at most one 3-cut and sufficiently many edges.

Approximation algorithms for the k-depots Hamiltonian path problem

Approximation algorithms for the k-depots Hamiltonian path problem

An improvement of spectral conditions for Hamilton-connected graphs

A conjecture by Erds and Hajnal [Ramsey-type theorems. Discrete Appl Math. 1989;25:37–52] pointed out that the structure of graphs with forbidden induced subgraphs is very different from general graphs. More precisely, they contain much larger cliques or independent sets. Inspired by this, we prove that non-Hamilton-connected graphs satisfying certain conditions contain a large clique. From the perspective of looking for a maximum clique, we present some sufficient conditions for a graph to be Hamilton-connected in terms of size, spectral radius and signless Laplacian spectral radius, respectively, which extend some corresponding results.

Some properties of open subset intersection graph of a topological space

In the recent paper, authors introduced a graph topological structure, called open subset intersection graph of a topological space ϒ(τ) on a finite set X. In this present paper, we continue the study of the graph ϒ(τ) on a finite set X. It is shown that, if τ is a discrete topology on X and |X| ≥ 3, then the graph ϒ(τ) is not bipartite and regular. If τ is a discrete topology on X with |X| = 3 then it is shown that the ϒ(τ) is an edge-transitive graph, Perfect graph, Eulerian and Hamiltonian graph . Moreover, we detremine exact value of the independence number, vertex connectivity and edge connectivity of the graph ϒ(τ).

High powers of Hamiltonian cycles in randomly augmented graphs

AbstractWe investigate the existence of powers of Hamiltonian cycles in graphs with large minimum degree to which some additional edges have been added in a random manner. For all integers , and , and for any we show that adding random edges to an ‐vertex graph with minimum degree at least yields, with probability close to one, the existence of the ‐th power of a Hamiltonian cycle. In particular, for and this implies that adding random edges to such a graph already ensures the ‐st power of a Hamiltonian cycle (proved independently by Nenadov and Trujić). In this instance and for several other choices of , and we can show that our result is asymptotically optimal.

Total Proper Connection and Graph Operations

A graph is said to be total-colored if all the edges and vertices of the graph are colored. A path in a total-colored graph is a total proper path if (i) any two adjacent edges on the path differ in color, (ii) any two internal adjacent vertices on the path differ in color, and (iii) any internal vertex of the path differs in color from its incident edges on the path. A total-colored graph is called total-proper connected if any two vertices of the graph are connected by a total proper path of the graph. For a connected graph [Formula: see text], the total proper connection number of [Formula: see text], denoted by [Formula: see text], is defined as the smallest number of colors required to make [Formula: see text] total-proper connected. In this paper, we study the total proper connection number for the graph operations. We find that [Formula: see text] is the total proper connection number for the join, the lexicographic product and the strong product of nearly all graphs. Besides, we study three kinds of graphs with one factor to be traceable for the Cartesian product as well as the permutation graphs of the star and traceable graphs. The values of the total proper connection number for these graphs are all [Formula: see text].

Efficient PTAS for the maximum traveling salesman problem in a metric space of fixed doubling dimension

The maximum traveling salesman problem (Max TSP) consists of finding a Hamiltonian cycle with the maximum total weight of the edges in a given complete weighted graph. This problem is APX-hard in the general metric case but admits polynomial-time approximation schemes in the geometric setting, when the edge weights are induced by a vector norm in fixed-dimensional real space. We propose the first approximation scheme for Max TSP in an arbitrary metric space of fixed doubling dimension. The proposed algorithm implements an efficient PTAS which, for any fixed \(\varepsilon \in (0,1)\), computes a \((1-\varepsilon )\)-approximate solution of the problem in cubic time. Additionally, we suggest a cubic-time algorithm which finds asymptotically optimal solutions of the metric Max TSP in fixed and sublogarithmic doubling dimensions.

Identification Conditions for the Solvability of NP-complete Problems for the Class of Pre-fractal Graphs

Modern network systems (unmanned aerial vehicles groups, social networks, network production chains, transport and logistics networks, communication networks, cryptocurrency networks) are distinguished by their multi-element nature and the dynamics of connections between its elements. A number of discrete problems on the construction of optimal substructures of network systems described in the form of various classes of graphs are NP-complete problems. In this case, the variability and dynamism of the structures of network systems leads to an "additional" complication of the search for solutions to discrete optimization problems. At the same time, for some subclasses of dynamical graphs, which are used to model the structures of network systems, conditions for the solvability of a number of NP-complete problems can be distinguished. This subclass of dynamic graphs includes pre-fractal graphs. The article investigates NP-complete problems on pre-fractal graphs: a Hamiltonian cycle, a skeleton with the maximum number of pendant vertices, a monochromatic triangle, a clique, an independent set. The conditions under which for some problems it is possible to obtain an answer about the existence and to construct polynomial (when fixing the number of seed vertices) algorithms for finding solutions are identified.

Oriented bipartite graphs and the Goldbach graph

In this paper, we study oriented bipartite graphs. In particular, we introduce “bitransitive” graphs. Several characterizations of bitransitive bitournaments are obtained. We show that bitransitive bitounaments are equivalent to acyclic bitournaments. As applications, we characterize acyclic bitournaments with Hamiltonian paths, determine the number of non-isomorphic acyclic bitournaments of a given order, and solve the graph-isomorphism problem in linear time for acyclic bitournaments. Next, we prove the well-known Caccetta-Häggkvist Conjecture for oriented bipartite graphs in some cases for which it is unsolved, in general, for oriented graphs. We also introduce the concept of undirected as well as oriented “odd-even” graphs. We characterize bipartite graphs and acyclic oriented bipartite graphs in terms of them. In fact, we show that any bipartite graph (acyclic oriented bipartite graph) can be represented by some odd-even graph (oriented odd-even graph). We obtain some conditions for connectedness of odd-even graphs. This study of odd-even graphs and their connectedness is motivated by a special family of odd-even graphs which we call “Goldbach graphs”. We show that the famous Goldbach's conjecture is equivalent to the connectedness of Goldbach graphs. Several other number theoretic conjectures (e.g., the twin prime conjecture) are related to various parameters of Goldbach graphs, motivating us to study the nature of vertex-degrees and independent sets of these graphs. Finally, we observe Hamiltonian properties of some odd-even graphs related to Goldbach graphs for a small number of vertices.

Hamiltonian cycles and paths in hypercubes with disjoint faulty edges

We consider hypercubes with pairwise disjoint faulty edges. An n-dimensional hypercube Qn is an undirected graph with 2n nodes, each labeled with a distinct binary string of length n. The parity of the vertex is 0 if the number of ones in its label is even, and is 1 if the number of ones is odd. Two vertices a and b are connected by an edge iff a and b differ in one position. If a and b differ in position i, then we say that the edge (a,b) goes in dimension i and we define the parity of the edge as the parity of the end with 0 on the position i. It was already known that Qn is not Hamiltonian if all edges going in one dimension and of the same parity are faulty.In this paper we show that if n≥4 then all other hypercubes are Hamiltonian. In other words, every cube Qn, with n≥4 and disjoint faulty edges is Hamiltonian if and only if for each dimension there are two healthy crossing edges of different parity.

New methods to attack the Buratti-Horak-Rosa conjecture

The conjecture, still widely open, posed by Marco Buratti, Peter Horak and Alex Rosa states that a list L of v−1 positive integers not exceeding ⌊v2⌋ is the list of edge-lengths of a suitable Hamiltonian path of the complete graph with vertex-set {0,1,…,v−1} if and only if, for every divisor d of v, the number of multiples of d appearing in L is at most v−d. In this paper we present new methods that are based on linear realizations and can be applied to prove the validity of this conjecture for a vast choice of lists. As example of their flexibility, we consider lists whose underlying set is one of the following: {x,y,x+y}, {1,2,3,4}, {1,2,4,…,2x}, {1,2,4,…,2x,2x+1}. We also consider lists with many consecutive elements.

Hamiltonian and long cycles in bipartite graphs with connectivity

Let G be a graph, ν(G) the order of G, κ(G) the connectivity of G and k a positive integer such that k≤(ν(G)−2)∕2. Then G is said to be k-extendable if it has a matching of size k and every matching of size k extends to a perfect matching of G. A graph G is Hamiltonian if it contains a Hamiltonian cycle. In this paper, we define a parameter c(G) and show some of its properties in bipartite graphs. Let G be a bipartite graph with bipartition (X,Y) such that |X|=|Y| and c(G)>0, and C a longest cycle in G. We show that 0<c(G)≤κ(G)−1, G is Hamiltonian if ν(G)≤2κ(G)+4c(G)−2, and |V(C)|≥6c(G) if ν(G)>6c(G). We show some of its corollaries in k-extendable, bipartite graphs and two conjectures in k-extendable graphs.

On the Hamilton laceability of double generalized Petersen graphs

A bipartite graph with bipartition A and B is said to be Hamilton-laceable if for any u∈A and v∈B there is a Hamilton path joining u and v. It is known that the double generalized Petersen graph DP(n,k) is Hamiltonian and is bipartite if and only if n is even. In this paper we show that the bipartite double generalized Petersen graph DP(n,k) is Hamilton-laceable for n≥4.

The Steiner cycle and path cover problem on interval graphs

The Steiner path problem is a common generalization of the Steiner tree and the Hamiltonian path problem, in which we have to decide if for a given graph there exists a path visiting a fixed set of terminals. In the Steiner cycle problem we look for a cycle visiting all terminals instead of a path. The Steiner path cover problem is an optimization variant of the Steiner path problem generalizing the path cover problem, in which one has to cover all terminals with a minimum number of paths. We study those problems for the special class of interval graphs. We present linear time algorithms for both the Steiner path cover problem and the Steiner cycle problem on interval graphs given as endpoint sorted lists. The main contribution is a lemma showing that backward steps to non-Steiner intervals are never necessary. Furthermore, we show how to integrate this modification to the deferred-query technique of Chang et al. to obtain the linear running times.

Pancyclicity in strong [formula omitted]-quasi-transitive digraphs of large diameter

A digraph is k-quasi-transitive if the existence of a directed path of length k from vertex u to vertex v implies that u and v are adjacent; this family is a generalization of both tournaments and transitive digraphs.Wang and Zhang proved that, when k is even, every strong k-quasi-transitive digraph with diameter at least k+2 contains a hamiltonian path, and asked whether under the same assumptions, a hamiltonian cycle could be found. In this work we answer this question in the affirmative, and moreover, we prove that not only hamiltonicity, but pancyclicity can be proved. As a byproduct of our proof, we derive a polynomial time algorithm to find a hamiltonian cycle in a k-quasi-transitive digraph with diameter at least k+2, for even values of k.