Edge-disjoint Hamiltonian Cycles Research Articles

Three Edge-Disjoint Hamiltonian Cycles in Folded Locally Twisted Cubes and Folded Crossed Cubes with Applications to All-to-All Broadcasting

All-to-all broadcasting means to distribute the exclusive message of each node in the network to all other nodes. It can be handled by rings, and a Hamiltonian cycle is a ring that visits each vertex exactly once. Multiple edge-disjoint Hamiltonian cycles, abbreviated as EDHCs, have two application advantages: (1) parallel data broadcast and (2) edge fault-tolerance in network communications. There are three edge-disjoint Hamiltonian cycles on n-dimensional locally twisted cubes and n-dimensional crossed cubes while n ≥ 6, respectively. Locally twisted cubes, crossed cubes, folded locally twisted cubes (denoted as FLTQn), and folded crossed cubes (denoted as FCQn) are among the hypercube-variant network. The topology of hypercube-variant network has more wealth than normal hypercubes in network properties. Then, the following results are presented in this paper: (1) Using the technique of edge exchange, three EDHCs are constructed in FLTQ5 and FCQ5, respectively. (2) According to the recursive structure of FLTQn and FCQn, there are three EDHCs in FLTQn and FCQn while n ≥ 6. (3) Considering that multiple faulty edges will occur randomly, the data broadcast performance of three EDHCs in FLTQn and FCQn is evaluated by simulation when 5 ≤ n ≤ 9.

Recursive definition and two edge-disjoint Hamiltonian cycles of bubble-sort star graphs

The bubble-sort star graph is an interconnection network for multiprocessor systems. Recursive structure is important for interconnection networks, since it can reduce the complex cases into simple cases, and it can keep good properties independent of dimensions. Many algorithms use the idea of recursive construction. Edge-disjoint Hamiltonian cycles not only provide the basis of all-to-all broadcasting algorithm for networks but also provide a replaceable Hamiltonian cycle for transmission when the other Hamiltonian cycle contains faulty edges in an interconnection network. In this paper, we propose the recursive definition of bubble-sort star graphs. Then as an application, we obtain two edge-disjoint Hamiltonian cycles of bubble-sort star graphs.

Symmetric property and edge-disjoint Hamiltonian cycles of the spined cube

The spined cube SQn, as a variant network of the hypercube Qn, was proposed in 2011 and has attracted much attention because of its smaller diameter. It is well-known that Qn is a Cayley graph. In the present paper, we show that SQn is an m-Cayley graph, that is its automorphism group has a semiregular subgroup acting on the vertices with m orbits, where m=4 when n≥6 and m=⌊n/2⌋ when n≤5. Consequently, it shows that an SQn with n≥6 can be partitioned into eight disjoint hypercubes of dimension n−3. As an application, it is proved that there exist two edge-disjoint Hamiltonian cycles in SQn when n≥4. Moreover, we prove that SQn is not vertex-transitive unless n≤3.

Decomposing hypergraphs into cycle factors

A famous result by Rödl, Ruciński, and Szemerédi guarantees a (tight) Hamilton cycle in k-uniform hypergraphs H on n vertices with minimum (k−1)-degree δk−1(H)⩾(1/2+o(1))n, thereby extending Dirac's result from graphs to hypergraphs. For graphs, much more is known; each graph on n vertices with δ(G)⩾(1/2+o(1))n contains (1−o(1))r edge-disjoint Hamilton cycles where r is the largest integer such that G contains a spanning 2r-regular subgraph, which is clearly asymptotically optimal. This was proved by Ferber, Krivelevich, and Sudakov answering a question raised by Kühn, Lapinskas, and Osthus.We extend this result to hypergraphs; every k-uniform hypergraph H on n vertices with δk−1(H)⩾(1/2+o(1))n contains (1−o(1))r edge-disjoint (tight) Hamilton cycles where r is the largest integer such that H contains a spanning subgraph with each vertex belonging to kr edges. In particular, this yields an asymptotic solution to a question of Glock, Kühn, and Osthus.In fact, our main result applies to approximately vertex-regular k-uniform hypergraphs with a weak quasirandom property and provides approximate decompositions into cycle factors without too short cycles.

Three edge-disjoint Hamiltonian cycles in crossed cubes with applications to fault-tolerant data broadcasting

Three edge-disjoint Hamiltonian cycles in crossed cubes with applications to fault-tolerant data broadcasting

Maximal sets of Hamilton cycles in complete multipartite graphs IV

Finding the values of μ for which there exists a maximal set of μ edge-disjoint Hamilton cycles in the complete multipartite graph Knp has been considered in papers for over 20 years. This paper finally settles the problem by finding such a set in the last remaining open case, namely where μ is as small as possible (so its existence was still in doubt) when n=3 and the number of parts, p, is 3 (mod 4).

Hitting Time of Edge Disjoint Hamilton Cycles in Random Subgraph Processes on Dense Base Graphs

Consider the random subgraph process on a base graph $G$ on $n$ vertices: a sequence $\lbrace G_t \rbrace _{t=0} ^{|E(G)|}$ of random subgraphs of $G$ obtained by choosing an ordering of the edges of $G$ uniformly at random, and by sequentially adding edges to $G_0$, the empty graph on the vertex set of $G$, according to the chosen ordering. We show that if $G$ has one of the following properties: 1. There is a positive constant $\varepsilon > 0$ such that $\delta (G) \geq \left( \frac{1}{2} + \varepsilon \right) n$; 2. There are some constants $\alpha, \beta >0$ such that every two disjoint subsets $U,W$ of size at least $\alpha n$ have at least $\beta |U||W|$ edges between them, and the minimum degree of $G$ is at least $(2\alpha + \beta )\cdot n$; or: 3. $G$ is an $(n,d,\lambda )$--graph, with $d\geq \frac{C\cdot n\cdot \log \log n}{\log n}$ and $\lambda \leq \frac{c\cdot d^2}{n}$ for some absolute constants $c,C>0$. then for a positive integer constant $k$ with high probability the hitting time of the property of containing $k$ edge disjoint Hamilton cycles is equal to the hitting time of having minimum degree at least $2k$. These results extend prior results by by Johansson and by Frieze and Krivelevich, and answer a question posed by Frieze.

An Algorithm to Construct Completely Independent Spanning Trees in Line Graphs

Abstract In the past few years, much importance and attention have been attached to completely independent spanning trees (CISTs). Many results, such as edge-disjoint Hamilton cycles, traceability, number of spanning trees, structural properties, topological indices, etc., have been obtained on line graphs, and researchers have applied the line graphs of some interconnection networks such as generalized hypercubes, augmented cubes, crossed cubes, etc., into data center networks, such as SWCube, AQLCube, BCDC, etc. At the meanwhile, few results of CISTs are reported on the line graphs. In this paper, we establish the relation of edge-disjoint spanning trees in an interconnection network $G$’ with its line graph $G$ by proposing a general algorithm for the first time. By this method, more CISTs can be obtained comparing with results in the literature. Then, the decrease of diameter is discussed and simulation experiments are shown on the line graphs of hypercubes.

Backtracking Algorithms for Constructing the Hamiltonian Decomposition of a 4-regular Multigraph

We consider a Hamiltonian decomposition problem of partitioning a regular graph into edge-disjoint Hamiltonian cycles. It is known that verifying vertex non-adjacency in the 1-skeleton of the symmetric and asymmetric traveling salesperson polytopes is an NP-complete problem. On the other hand, a suffcient condition for two vertices to be non-adjacent can be formulated as a combinatorial problem of finding a Hamiltonian decomposition of a 4-regular multigraph. We present two backtracking algorithms for verifying vertex non-adjacency in the 1-skeleton of the traveling salesperson polytope and constructing a Hamiltonian decomposition: an algorithm based on a simple path extension and an algorithm based on the chain edge fixing procedure. Based on the results of the computational experiments for undirected multigraphs, both backtracking algorithms lost to the known heuristic general variable neighborhood search algorithm. However, for directed multigraphs, the algorithm based on chain fixing of edges showed comparable results with heuristics on instances with existing solutions, and better results on instances of the problem where the Hamiltonian decomposition does not exist.

Hamiltonian decomposition and verifying vertex adjacency in 1-skeleton of the traveling salesperson polytope by variable neighborhood search

We consider a Hamiltonian decomposition problem of partitioning a regular graph into edge-disjoint Hamiltonian cycles. A sufficient condition for vertex adjacency in the 1-skeleton of the traveling salesperson polytope can be formulated as the Hamiltonian decomposition problem in a 4-regular multigraph. We introduce a heuristic general variable neighborhood search algorithm for this problem based on finding a vertex-disjoint cycle cover of the multigraph through reduction to perfect matching and several cycle merging operations. The algorithm has a one-sided error: the answer "not adjacent" is always correct, and was tested on random directed and undirected Hamiltonian cycles and on pyramidal tours.

Maximal sets of Hamilton cycles in [formula omitted

Let Knr;λ1,λ2 be the r-partite multigraph in which each part has size n, where two vertices in the same part or different parts are joined by exactly λ1 edges or λ2 edges, respectively. It is proved that there exists a maximal set of t edge-disjoint Hamilton cycles in Knr;λ1,λ2 for λ2n⌊r+34⌋≤t≤min{⌊λ2n2(r−1)2⌋,⌊λ1(n−1)+λ2n(r−1)2⌋}, the upper bound being best possible. The results proved make use of the method of amalgamations.

Edge-disjoint Hamiltonian cycles of balanced hypercubes

The balanced hypercube, a variant of the hypercube, was proposed as a novel interconnection network of parallel systems. It is known that the balanced hypercube is Hamiltonian. The existence of two edge-disjoint Hamiltonian cycles in balanced hypercubes has remained unknown. In this paper, we prove that the balanced hypercube BHn with n≥2 contains two edge-disjoint Hamiltonian cycles. Further, a linear time algorithm is provided to construct two edge-disjoint Hamiltonian cycles in the balanced hypercube BHn with n≥2.

Optimal Bounds for Disjoint Hamilton Cycles in Star Graphs

In interconnection network topologies, the [Formula: see text]-dimensional star graph [Formula: see text] has [Formula: see text] vertices corresponding to permutations [Formula: see text] of [Formula: see text] symbols [Formula: see text] and edges which exchange the positions of the first symbol [Formula: see text] with any one of the other symbols. The star graph compares favorably with the familiar [Formula: see text]-cube on degree, diameter and a number of other parameters. A desirable property which has not been fully evaluated in star graphs is the presence of multiple edge-disjoint Hamilton cycles which are important for fault-tolerance. The only known method for producing multiple edge-disjoint Hamilton cycles in [Formula: see text] has been to label the edges in a certain way and then take images of a known base 2-labelled Hamilton cycle under different automorphisms that map labels consistently. However, optimal bounds for producing edge-disjoint Hamilton cycles in this way, and whether Hamilton decompositions can be produced, are not known for any [Formula: see text] other than for the case of [Formula: see text] which does provide a Hamilton decomposition. In this paper we show that, for all n, not more than [Formula: see text], where [Formula: see text] is Euler’s totient function, edge-disjoint Hamilton cycles can be produced by such automorphisms. Thus, for non-prime [Formula: see text], a Hamilton decomposition cannot be produced. We show that the [Formula: see text] upper bound can be achieved for all even [Formula: see text]. In particular, if [Formula: see text] is a power of 2, [Formula: see text] has a Hamilton decomposable spanning subgraph comprising more than half of the edges of [Formula: see text]. Our results produce a better than twofold improvement on the known bounds for any kind of edge-disjoint Hamilton cycles in [Formula: see text]-dimensional star graphs for general [Formula: see text].

Packing Hamilton Cycles Online

It is known that w.h.p. the hitting time τ2σ for the random graph process to have minimum degree 2σ coincides with the hitting time for σ edge-disjoint Hamilton cycles [4, 9, 13]. In this paper we prove an online version of this property. We show that, for a fixed integer σ ⩾ 2, if random edges of Kn are presented one by one then w.h.p. it is possible to colour the edges online with σ colours so that at time τ2σ each colour class is Hamiltonian.

Packing Loose Hamilton Cycles

A subsetCof edges in ak-uniform hypergraphHis aloose Hamilton cycleifCcovers all the vertices ofHand there exists a cyclic ordering of these vertices such that the edges inCare segments of that order and such that every two consecutive edges share exactly one vertex. The binomial randomk-uniform hypergraphHkn,phas vertex set [n] and an edge setEobtained by adding eachk-tuplee∈ ($\binom{[n]}{k}$) toEwith probabilityp, independently at random.Here we consider the problem of finding edge-disjoint loose Hamilton cycles covering all buto(|E|) edges, referred to as thepacking problem. While it is known that the threshold probability of the appearance of a loose Hamilton cycle inHkn,pis$p=\Theta\biggl(\frac{\log n}{n^{k-1}}\biggr),$the best known bounds for the packing problem are aroundp= polylog(n)/n. Here we make substantial progress and prove the following asymptotically (up to a polylog(n) factor) best possible result: forp≥ logCn/nk−1, a randomk-uniform hypergraphHkn,pwith high probability contains$N:=(1-o(1))\frac{\binom{n}{k}p}{n/(k-1)}$edge-disjoint loose Hamilton cycles.Our proof utilizes and modifies the idea of ‘online sprinkling’ recently introduced by Vu and the first author.

Higher dimensional Eisenstein–Jacobi networks

An efficient interconnection topology called Eisenstein–Jacobi (EJ) network has been proposed in Martínez et al. (2008). In this paper this concept is generalized to higher dimensions. Important properties such as distance distribution and the decomposition of higher dimensional EJ networks into edge-disjoint Hamiltonian cycles are explored in this paper. In addition, an optimal shortest path routing algorithm and a one-to-all broadcast algorithm for higher dimensional EJ networks are given. Further, we give comparisons between higher EJ networks and Generalized Hypercube (GHC) networks and we show that higher EJ networks cost less and have more nodes than GHC networks.

Proof of the 1-factorization and Hamilton Decomposition Conjectures

We prove the following results (via a unified approach) for all sufficiently large n: (i) [1 -factorization conjecture] Suppose that n is even and D ≥ 2⌈n/4⌉ − 1. Then every D-regular graph G on n vertices has a decomposition into perfect matchings. Equivalently, χ′(G) = D. (ii) [Hamilton decomposition conjecture] Suppose that D ≥ ⌊n/2⌋. Then every D-regular graph G on n vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) [Optimal packings of Hamilton cycles] Suppose that G is a graph on n vertices with minimum degree δ ≥ n/2. Then G contains at least (n − 2)/8 edge-disjoint Hamilton cycles. According to Dirac, (i) was first raised in the 1950’s. (ii) and (iii) answer questions of Nash-Williams from 1970. All of the above bounds are best possible.

Higher Dimensional Gaussian Networks

Gaussian interconnection networks have been introduced as a useful alternative to the classical toroidal networks, and in this paper this concept is generalized to higher dimensions. We also explore many important properties of this new topology, including distance distribution and the decomposition of higher dimensional Gaussian networks into edge-disjoint tori and Hamiltonian cycles. In addition, an optimal shortest path routing algorithm and a one-to-all broadcast algorithm for higher dimensional Gaussian networks are given. Simulation results show that the routing algorithm proposed for higher dimensional Gaussian networks outperforms the routing algorithm of the corresponding torus network of the same node-degree and the same number of nodes.

Counting and packing Hamilton cycles in dense graphs and oriented graphs

We present a general method for counting and packing Hamilton cycles in dense graphs and oriented graphs, based on permanent estimates. We utilize this approach to prove several extremal results. In particular, we show that every nearly cn-regular oriented graph on n vertices with c>3/8 contains (cn/e)n(1+o(1))n directed Hamilton cycles. This is an extension of a result of Cuckler, who settled an old conjecture of Thomassen about the number of Hamilton cycles in regular tournaments. We also prove that every graph G on n vertices of minimum degree at least (1/2+o(1))n contains at least (1−o(1))regeven(G)/2 edge-disjoint Hamilton cycles, where regeven(G) is the maximum even degree of a spanning regular subgraph of G. This establishes an approximate version of a conjecture of Kühn, Lapinskas and Osthus.

Edge Disjoint Hamiltonian Cycles in Highly Connected Tournaments

Thomassen conjectured that there is a function f(k) such that every strongly f(k)-connected tournament contains k edge-disjoint Hamiltonian cycles. This conjecture was recently proved by Kuhn, Lapinskas, Osthus, and Patel who showed that f(k) ≤ O(k 2 (logk) 2 ) and conjectured that there is a constant C such that f(k) ≤ Ck 2 . We prove this conjecture. As a second application of our methods we answer a question of Thomassen about spanning linkages in highly connected tournaments.