Edge-disjoint Cycles Research Articles

Decompositions of complete multigraphs into cycles of varying lengths

We establish necessary and sufficient conditions for the existence of a decomposition of a complete multigraph into edge-disjoint cycles of specified lengths, or into edge-disjoint cycles of specified lengths and a perfect matching.

Cycle multiplicity of some total graphs

Cycle multiplicity of a graph G is the maximum number of edge disjoint cycles in G. Simoes Pereira gave the formula of cycle multiplicity for line and total graph of forests. Recently, Akbar Ali determined cycle multiplicity of total graph for Cn and K1, n. In this paper, we determine the cycle multiplicity of some Cartesian product graphs, the total graphs of complete graph, complete bipartite graph and unicycle graph, which generalize the results of Akbar Ali and Panayappan (2010) in [1].

A Dynamic Programming Approach for the Max-Min Cycle Packing Problem in Even Graphs

Let be an undirected graph. The maximum cycle packing problem in G then is to find a collection of edge-disjoint cycles Ciin G such that s is maximum. In general, the maximum cycle packing problem is NP-hard. In this paper, it is shown for even graphs that if such a collection satisfies the condition that it minimizes the quantityon the set of all edge-disjoint cycle collections, then it is a maximum cycle packing. The paper shows that the determination of such a packing can be solved by a dynamic programming approach. For its solution, an-shortest path procedure on an appropriate acyclic networkis presented. It uses a particular monotonous node potential.

Skew-rank of an oriented graph with edge-disjoint cycles

An oriented graph is a digraph without loops and multiple arcs, where is called the underlying graph of . Let denote the skew-adjacency matrix of . The rank of is called the skew-rank of , denoted by , which is even since is skew symmetric. Recently, Qu and Yu proved that for an oriented bicyclic graph with pendant vertices and with two edge-disjoint cycles of size and . In this paper, we extend this result to a more general case. It is proved that if is a connected oriented graph with pairwise edge-disjoint cycles of size . Moreover, the extremal graphs attaining the lower bound are characterized.

Approximation Algorithms for Min-Max Cycle Cover Problems

As a fundamental optimization problem, the vehicle routing problem has wide application backgrounds and has been paid lots of attentions in past decades. In this paper we study its applications in data gathering and wireless energy charging for wireless sensor networks, by devising improved approximation algorithms for it and its variants. The key ingredients in the algorithm design include exploiting the combinatorial properties of the problems and making use of tree decomposition and minimum weighted maximum matching techniques. Specifically, given a metric complete graph $G$ and an integer $k>0$ , we consider rootless, uncapacitated rooted, and capacitated rooted min-max cycle cover problems in $G$ with an aim to find $k$ rootless (or rooted) edge-disjoint cycles covering the vertices in $V$ such that the maximum cycle weight among the $k$ cycles is minimized. For each of the mentioned problems, we develop an improved approximate solution. That is, for the rootless min-max cycle cover problem, we develop a $(5{ 1\over 3} +\epsilon)$ -approximation algorithm; for the uncapacitated rooted min-max cycle cover problem, we devise a $(6{ 1\over 3} +\epsilon)$ -approximation algorithm; and for the capacitated rooted min-max cycle cover problem, we propose a $(7+\epsilon)$ -approximation algorithm. These algorithms improve the best existing approximation ratios of the corresponding problems $6+\epsilon$ , $7+\epsilon$ , and $13+\epsilon$ , respectively, where $\epsilon$ is a constant with $0 . We finally evaluate the performance of the proposed algorithms through experimental simulations. Experimental results show that the actual approximation ratios delivered by the proposed algorithms are always no more than 2, much better than their analytical counterparts.

English

This paper addresses upper and lower bounds for the cardinality of a maximum vertex-/edge-disjoint cycle packing in a polyhedral graph G. Bounds on the cardinality of such packings are provided, that depend on the size, the order or the number of faces of G, respectively. Polyhedral graphs are constructed, that attain these bounds.

Color the cycles

The cycles of length k in a complete graph on n vertices are colored in such a way that edge-disjoint cycles get distinct colors. The minimum number of colors is asymptotically determined.

Bose–Hubbard model on two-dimensional line graphs

We construct a basis for the many-particle ground states of the positive hopping Bose–Hubbard model on line graphs of finite 2-connected planar bipartite graphs at sufficiently low filling factors. The particles in these states are localized on non-intersecting vertex-disjoint cycles of the line graph which correspond to non-intersecting edge-disjoint cycles of the original graph. The construction works up to a critical filling factor at which the cycles are close-packed.

Minimum Cycle Covers of Butterfly and Benes Networks

Butterfly network is the most popular bounded-degree derivative of the hypercube network. The benes network consists of back-to-back butterflies. In this paper, we obtain the minimum vertex-disjoint cycle cover number for the odd dimensional butterfly networks and prove that it is not possible to find the same for the even dimensional butterfly networks and benes networks. Further we obtain the minimum edge-disjoint cycle cover number for butterfly networks.

Cycle decompositions: From graphs to continua

We generalise a fundamental graph-theoretical fact, stating that every element of the cycle space of a graph is a sum of edge-disjoint cycles, to arbitrary continua. To achieve this we replace graph cycles by topological circles, and replace the cycle space of a graph by a new homology group for continua which is a quotient of the first singular homology group H1. This homology seems to be particularly apt for studying spaces with infinitely generated H1, e.g. infinite graphs or fractals.

The Signless Laplacian Spectral Radius of Unicyclic and Bicyclic Graphs with a Given Girth

Let $\mathcal{U}(n,g)$ and $\mathcal{B}(n,g)$ be the set of unicyclic graphs and bicyclic graphs on $n$ vertices with girth $g$, respectively. Let $\mathcal{B}_{1}(n,g)$ be the subclass of $\mathcal{B}(n,g)$ consisting of all bicyclic graphs with two edge-disjoint cycles and $\mathcal{B}_{2}(n,g)=\mathcal{B}(n,g)\backslash\mathcal{B}_{1}(n,g)$. This paper determines the unique graph with the maximal signless Laplacian spectral radius among all graphs in $\mathcal{U}(n,g)$ and $\mathcal{B}(n,g)$, respectively. Furthermore, an upper bound of the signless Laplacian spectral radius and the extremal graph for $\mathcal{B}(n,g)$ are also given.

Cycle multiplicity of total graph of Cn, Pn, and K1,n

Cycle multiplicity of a graph G is the maximum number of edge disjoint cycles in G. In this paper, we find the cycle multiplicity of total graph of cycles C n , paths P n , and star graph K 1,n respectively. Keywords: cycle multiplicity, total graph, cycle, path, star graph.

Packing disjoint cycles over vertex cuts

For a graph G , let ν ( G ) and ν ′ ( G ) denote the maximum cardinalities of packings of vertex-disjoint and edge-disjoint cycles of G , respectively. We study the interplay of these two parameters and vertex cuts in graphs. If G is a graph whose vertex set can be partitioned into three non-empty sets S , V 1 , and V 2 such that there is no edge between V 1 and V 2 , and k = | S | , then our results imply that ν ( G ) is uniquely determined by the values ν ( H ) for at most 2 k + 1 k ! 2 graphs H of order at most max { | V 1 | , | V 2 | } + k , and ν ′ ( G ) is uniquely determined by the values ν ′ ( H ) for at most 2 k 2 + 1 graphs H of order at most max { | V 1 | , | V 2 | } + k .

An asymptotic solution to the cycle decomposition problem for complete graphs

Let m 1 , m 2 , … , m t be a list of integers. It is shown that there exists an integer N such that for all n ⩾ N , the complete graph of order n can be decomposed into edge-disjoint cycles of lengths m 1 , m 2 , … , m t if and only if n is odd, 3 ⩽ m i ⩽ n for i = 1 , 2 , … , t , and m 1 + m 2 + ⋯ + m t = ( n 2 ) . In 1981, Alspach conjectured that this result holds for all n, and that a corresponding result also holds for decompositions of complete graphs of even order into cycles and a perfect matching.

Approximability of Packing Disjoint Cycles

Given a graph G, the edge-disjoint cycle packing problem is to find the largest set of cycles of which no two share an edge. For undirected graphs, the best known approximation algorithm has ratio $O(\sqrt{\log n})$ (Krivelevich et al. in ACM Trans. Algorithms, 2009, to appear). In fact, they proved the same upper bound for the integrality gap of this problem by presenting a simple greedy algorithm. Here we show that this is almost best possible. By modifying integrality gap and hardness results for the edge-disjoint paths problem (Andrews et al. in Proc. of 46th IEEE FOCS, pp. 226–244, 2005; Chuzhoy and Khanna in New hardness results for undirected edge disjoint paths. Manuscript, 2005), we show that the undirected edge-disjoint cycle packing problem is quasi-NP-hard to approximate within ratio of $O(\log^{\frac{1}{2}-\epsilon}n)$ for any constant e>0. The same result holds for the problem of packing vertex-disjoint cycles.

Packing edge-disjoint cycles in graphs and the cyclomatic number

For a graph G let μ ( G ) denote the cyclomatic number and let ν ( G ) denote the maximum number of edge-disjoint cycles of G . We prove that for every k ≥ 0 there is a finite set P ( k ) such that every 2-connected graph G for which μ ( G ) − ν ( G ) = k arises by applying a simple extension rule to a graph in P ( k ) . Furthermore, we determine P ( k ) for k ≤ 2 exactly.

The Laplacian spectral radius of bicyclic graphs with a given girth

Let ℬ ( n , g ) be the class of bicyclic graphs on n vertices with girth g . Let ℬ 1 ( n , g ) be the subclass of ℬ ( n , g ) consisting of all bicyclic graphs with two edge-disjoint cycles and ℬ 2 ( n , g ) = ℬ ( n , g ) ∖ ℬ 1 ( n , g ) . This paper determines the unique graph with the maximal Laplacian spectral radius among all graphs in ℬ 1 ( n , g ) and ℬ 2 ( n , g ) , respectively. Furthermore, the upper bound of the Laplacian spectral radius and the extremal graph for ℬ ( n , g ) are also obtained.

On packing shortest cycles in graphs

We study the problems to find a maximum packing of shortest edge-disjoint cycles in a graph of given girth g ( g-ESCP) and its vertex-disjoint analogue g-VSCP. In the case g = 3 , Caprara and Rizzi (2001) have shown that g-ESCP can be solved in polynomial time for graphs with maximum degree 4, but is APX-hard for graphs with maximum degree 5, while g-VSCP can be solved in polynomial time for graphs with maximum degree 3, but is APX-hard for graphs with maximum degree 4. For g ∈ { 4 , 5 } , we show that both problems allow polynomial time algorithms for instances with maximum degree 3, but are APX-hard for instances with maximum degree 4. For each g ⩾ 6 , both problems are APX-hard already for graphs with maximum degree 3.

Reductions in Computational Complexity Using Clifford Algebras

A number of combinatorial problems are treated using properties of abelian null-square-generated and idempotent-generated subalgebras of Clifford algebras. For example, the problem of deciding whether or not a graph contains a Hamiltonian cycle is known to be NP-complete. By considering entries of Λk, where Λ is an appropriate nilpotent adjacency matrix, the k-cycles in any finite graph are recovered. Within the algebra context (i.e., considering the number of multiplications performed within the algebra), these problems are reduced to matrix multiplication, which is in complexity class P. The Hamiltonian cycle problem is one of many problems moved from classes NP-complete and #P-complete to class P in this context. Other problems considered include the set covering problem, counting the edge-disjoint cycle decompositions of a finite graph, computing the permanent of an arbitrary matrix, computing the girth and circumference of a graph, and finding the longest path in a graph.

Packing cycles in complete graphs

We introduce a new technique for packing pairwise edge-disjoint cycles of specified lengths in complete graphs and use it to prove several results. Firstly, we prove the existence of dense packings of the complete graph with pairwise edge-disjoint cycles of arbitrary specified lengths. We then use this result to prove the existence of decompositions of the complete graph of odd order into pairwise edge-disjoint cycles for a large family of lists of specified cycle lengths. Finally, we construct new maximum packings of the complete graph with pairwise edge-disjoint cycles of uniform length.