Vertex-disjoint Cycles Research Articles

Two-disjoint-cycle-cover edge/vertex bipancyclicity of star graphs

A bipartite graph G is two-disjoint-cycle-cover edge [r1,r2]-bipancyclic if, for any vertex-disjoint edges uv and xy in G and any even integer ℓ satisfying r1⩽ℓ⩽r2, there exist vertex-disjoint cycles C1 and C2 such that |V(C1)|=ℓ, |V(C2)|=|V(G)|−ℓ, uv∈E(C1) and xy∈E(C2). In this paper, we prove that the n-star graph Sn is two-disjoint-cycle-cover edge [6,n!2]-bipancyclic for n⩾5, and thus it is two-disjoint-cycle-cover vertex [6,n!2]-bipancyclic for n⩾5. Additionally, it is examined that Sn is two-disjoint-cycle-cover [6,n!2]-bipancyclic for n⩾4.

Approximations for the Steiner Multicycle problem

The Steiner Multicycle problem consists in, given a complete graph, a weight function on its edges, and a collection of pairwise disjoint non-unitary sets called terminal sets, finding a minimum weight collection of vertex-disjoint cycles in the graph such that, for every terminal set, all of its vertices are in a same cycle of the collection. This problem generalizes the Traveling Salesman problem and, therefore, is hard to approximate in general. On the practical side, it models a collaborative less-than-truckload problem with pickup and delivery locations. Using an algorithm for the Survivable Network Design problem and T-joins, we obtain a 3-approximation for the metric case, improving on the previous best 4-approximation. Furthermore, we present an (11/9)-approximation for the particular case of the Steiner Multicycle in which each edge weight is 1 or 2. This algorithm can be adapted to obtain a (7/6)-approximation when every terminal set contains at least four vertices. Finally, we devise an O(lg⁡n)-approximation algorithm for the asymmetric version of the problem.

On extremal factors of de Bruijn-like graphs

In 1972, Mykkeltveit confirmed Golomb’s conjecture on the de Bruijn graphs: he proved that the pure cycling register rule yields the maximum number of vertex-disjoint cycles in de Bruijn graphs. We show that this result encompasses the tensor product of the de Bruijn graph for strings of length n with a simple cycle of size k, when n divides k or vice versa. Furthermore, we give counting formulas for an array of cycling register rules, which includes Golomb’s well-studied linear register rules.

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.

A solution to the conjecture of vertex disjoint cycles in graphs with partial degree

A solution to the conjecture of vertex disjoint cycles in graphs with partial degree

Embedding Spanning Disjoint Cycles in Hypercube Networks with Prescribed Edges in Each Cycle

One of the important issues in evaluating an interconnection network is to study the hamiltonian cycle embedding problems. A graph G is spanning k-edge-cyclable if for any k independent edges e1,e2,…,ek of G, there exist k vertex-disjoint cycles C1,C2,…,Ck in G such that V(C1)∪V(C2)∪⋯∪V(Ck)=V(G) and ei∈E(Ci) for all 1≤i≤k. According to the definition, the problem of finding hamiltonian cycle focuses on k=1. The notion of spanning edge-cyclability can be applied to the problem of identifying faulty links and other related issues in interconnection networks. In this paper, we prove that the n-dimensional hypercube Qn is spanning k-edge-cyclable for 1≤k≤n−1 and n≥2. This is the best possible result, in the sense that the n-dimensional hypercube Qn is not spanning n-edge-cyclable.

Vertex-Disjoint Cycles of Different Lengths in Local Tournaments

Vertex-Disjoint Cycles of Different Lengths in Local Tournaments

On the existence of k $k$‐cycle semiframes for even k $k$

AbstractA ‐semiframe of type is a ‐group divisible design of type in which is the vertex set, is the group set, and the set of ‐cycles can be written as a disjoint union where is partitioned into parallel classes on and is partitioned into holey parallel classes, each parallel class or holey parallel class being a set of vertex disjoint cycles whose vertex sets partition or for some . In this paper, we almost completely solve the existence of a ‐semiframe of type for all and a ‐semiframe of type for all and .

Rooted Prism-Minors and Disjoint Cycles Containing a Specified Edge

Dirac and Lovász independently characterized the $3$-connected graphs with no pair of vertex-disjoint cycles. Equivalently, they characterized all $3$-connected graphs with no prism-minors. In this paper, we completely characterize the $3$-connected graphs with no edge that is contained in the union of a pair of vertex-disjoint cycles. As applications, we answer the analogous questions for edge-disjoint cycles and for $4$-connected graphs and we completely characterize the $3$-connected graphs with no prism-minor using a specified edge.

Approximation Algorithms for the Maximum-Weight Cycle/Path Packing Problems

Given an undirected complete graph [Formula: see text] on [Formula: see text] vertices with a non-negative weight function on [Formula: see text], the maximum-weight [Formula: see text]-cycle ([Formula: see text]-path) packing problem aims to compute a set of [Formula: see text] vertex-disjoint cycles (paths) in [Formula: see text] containing [Formula: see text] vertices so that the total weight of the edges in these [Formula: see text] cycles (paths) is maximized. For the maximum-weight [Formula: see text]-cycle packing problem, we develop an algorithm achieving an approximation ratio of [Formula: see text], where [Formula: see text] is the approximation ratio for the maximum traveling salesman problem. For the case [Formula: see text], we design a better [Formula: see text]-approximation algorithm. When the weights of edges obey the triangle inequality, we propose a [Formula: see text]-approximation algorithm and a [Formula: see text]-approximation algorithm for the maximum-weight [Formula: see text]-cycle packing problem with [Formula: see text] and [Formula: see text], respectively. For the maximum-weight [Formula: see text]-path packing problem with [Formula: see text] (or [Formula: see text]) with the triangle inequality, we devise an algorithm with approximation ratio [Formula: see text] and give a tight example.

How to build a pillar: A proof of Thomassen's conjecture

Carsten Thomassen in 1989 conjectured that if a graph has minimum degree much more than the number of atoms in the universe (δ(G)≥101010), then it contains a pillar, which is a graph that consists of two vertex-disjoint cycles of the same length, s say, along with s vertex-disjoint paths of the same length3 which connect matching vertices in order around the cycles. Despite the simplicity of the structure of pillars and various developments of powerful embedding methods for paths and cycles in the past three decades, this innocent looking conjecture has seen no progress to date. In this paper, we give a proof of this conjecture by building a pillar (algorithmically) in sublinear expanders.

Two disjoint cycles in digraphs

AbstractBermond and Thomassen conjectured that every digraph with minimum outdegree at least contains vertex disjoint cycles. So far the conjecture was verified for . Here we generalise the question asking for all outdegree sequences which force vertex disjoint cycles and give the full answer for .

Two-disjoint-cycle-cover vertex bipancyclicity of bipartite hypercube-like networks

Let r1,r2 be two integers such that r2≥r1≥0. A bipartite graph G is two-disjoint-cycle-cover vertex [r1,r2]-bipancyclic (2-DCC vertex [r1,r2]-bipancyclic for short) if for any two vertices u,v∈V(G) and any even integer ℓ satisfying r1≤ℓ≤r2, there exist two vertex-disjoint cycles C1 and C2 in G with |V(C1)|=ℓ and |V(C2)|=|V(G)|−ℓ such that u∈V(C1) and v∈V(C2). In this paper, we study the 2-DCC vertex bipancyclicity of the n-dimensional bipartite hypercube-like network, which is one class of hypercube-generalized networks. As a consequence, we show that an n-dimensional bipartite hypercube-like network is 2-DCC vertex [4,2n−1]-bipancyclic for n≥3. In particular, it provides an application that n-dimensional hypercube and bicube are also 2-DCC vertex [4,2n−1]-bipancyclic for n≥3.

Disjoint cycles in graphs with restricted independence number

In 1963, Corrádi and Hajnal proved that every graph with at least 3 k vertices and minimum degree at least 2 k contains a collection of k vertex-disjoint cycles. The sharpness examples for this theorem were characterized by Kierstead, Kostochka, and Yeager in 2017 and one consequence of this characterization is that when k ≥ 3 , every graph with n ≥ 3 k vertices, minimum degree at least 2 k − 1 , and independence number at most n − 2 k − 1 has k vertex-disjoint cycles. We extend this result by showing that there exists β > 0 and t 0 such that for every t ≥ t 0 , k ≥ 25 t and n ≥ 4 k + t , every graph on n vertices with minimum degree at least 2 k − t and independence number at most n − 2 k − t + β t log t contains a collection of k vertex-disjoint cycles. We also show that the condition on the independence number is sharp up to the constant β .

Packing Directed Cycles Quarter- and Half-Integrally

The celebrated Erdős-Pósa theorem states that every undirected graph that does not admit a family of k vertex-disjoint cycles contains a feedback vertex set (a set of vertices hitting all cycles in the graph) of size $$\cal{O}(k\log k)$$ . The analogous result for directed graphs has been proven by Reed, Robertson, Seymour, and Thomas, but their proof yields a nonelementary dependency of the size of the feedback vertex set on the size of vertex-disjoint cycle packing. We show that we can obtain a polynomial bound if we relax the disjointness condition. More precisely, we show that if in a directed graph G there is no family of k cycles such that every vertex of G is in at most two (resp. four) of the cycles, then there exists a feedback vertex set in G of size $$\cal{O}(k^{6})$$ (resp. $$\cal{O}(k^{4})$$ ). We show also variants of the above statements for butterfly minor models of any strongly connected digraph that is a minor of a directed cylindrical grid and for quarter-integral packings of subgraphs of high directed treewidth.

Distance spectrum, 1-factor and vertex-disjoint cycles

Motivated by recent progresses on study of factors and spectral extremal graph theory, in this paper, we focus on distance spectral extrema of k-factors in bipartite graphs for small k. First of all, we aim to investigate the existence of 1-factor in a bipartite graph with given minimum degree in terms of its distance spectral radius, which generalize and improve the previous result in [37]. Then we determine the extremal graph attaining the minimum distance spectral radius among all bipartite graphs with a unique perfect matching. Notice that a 2-factor consists of some vertex-disjoint cycles. As a beginning, we naturally consider some propositions of vertex-disjoint cycles and give a sufficient condition for the existence of two vertex-disjoint cycles in a bipartite graph with respect to the distance spectral radius.

Domination, Independence and Fibonacci Numbers in Graphs Containing Disjoint Cycles

Graph theory is a delightful playground for the exploration of proof techniques in discrete mathematics and its results have applications in many areas of the computing, social, and natural sciences. The fastest growing area within graph theory is the study of domination and Independence numbers. Domination number is the cardinality of a minimum dominating set of a graph. Independence number is the maximal cardinality of an independent set of vertices of a graph. The concept of Fibonacci numbers of graphs was first introduced by Prodinger and Tichy in 1982. The Fibonacci numbers of a graph is the number of independent vertex subsets. In this paper, introduce the identities of domination, independence and Fibonacci numbers of graphs containing vertex-disjoint cycles and edge-disjoint cycles.

Length-constrained cycle partition with an application to UAV routing*

This article discusses the Length-Constrained Cycle Partition Problem (LCCP), which constitutes a new generalization of the Travelling Salesperson Problem (TSP). Apart from nonnegative edge weights, the undirected graph in LCCP features a nonnegative critical length parameter for each vertex. A cycle partition, i.e. a vertex-disjoint cycle cover, is a feasible solution for LCCP if the length of each cycle is not greater than the critical length of each vertex contained in it. The goal is to find a feasible partition having a minimum number of cycles. Besides analyzing theoretical properties and developing preprocessing techniques, we propose an elaborate heuristic algorithm that produces solutions of good quality even for large-size instances. Moreover, we present two exact mixed-integer programming formulations (MIPs) for LCCP, which are inspired by well-known modeling approaches for TSP. Further, we introduce the concept of conflict hypergraphs, whose cliques yield valid constraints for the MIP models. We conclude with a discussion on computational experiments that we conducted using (A)TSPLIB-based problem instances. As a motivating example application, we describe a routing problem where a fleet of uncrewed aerial vehicles (UAVs) must patrol a given set of areas.

Vertex-disjoint cycles of different lengths in multipartite tournaments

We show in this paper that every strong k-partite tournament D=(V,A) with k≥3 and minimum out-degree 3, except the digraphs D73 and D83 which are defined in the introduction of this paper, contains two vertex-disjoint directed cycles of different lengths.

Hypergraph Turán Numbers of Vertex Disjoint Cycles

Hypergraph Turán Numbers of Vertex Disjoint Cycles