Hamiltonian Cycles In Hypercubes Research Articles

A kind of matchings extend to hamiltonian cycles in hypercubes

Ruskey and Savage asked the following question: Does every matching in $Q_{n}$ for $n\geq2$ extend to a Hamiltonian cycle of $Q_{n}$? Kreweras conjectured that every perfect matching of $Q_n$ for $n\geq2$ can be extended to a Hamiltonian cycle of $Q_n$. Fink confirmed the conjecture. An edge in $Q_n$ is an edge of direction $i$ if its endpoints differ in the $i$th position. So all the edges of $Q_n$ can be divided into $n$ directions, i.e., edges of direction $1$, $\ldots$, edges of direction $n$. The set of all edges of direction $i$ of $Q_n$ is denoted by $E_i$. In this paper, we obtain the following result. For $n\geq6$, let $M$ be a matching in $Q_n$ with $|M|<10\times2^{n-5}$. If $M$ contains edges in at most 5 directions, then $M$ can be extended to a Hamiltonian cycle of $Q_n$.

Matchings extend to Hamiltonian cycles in hypercubes with faulty edges

We consider the problem of existence of a Hamiltonian cycle containing a matching and avoiding some edges in an n-cube Qn; and obtain the following results. Let n ⩾ 3; M ⊂ E(Qn); and F ⊂ E(Qn)M with 1 ⩽ |F| ⩽ 2n − 4 − |M|: If M is a matching and every vertex is incident with at least two edges in the graph Qn − F; then all edges of M lie on a Hamiltonian cycle in Qn − F: Moreover, if |M| = 1 or |M| = 2; then the upper bound of number of faulty edges tolerated is sharp. Our results generalize the well-known result for |M| = 1

Perfect matchings extend to two or more hamiltonian cycles in hypercubes

Kreweras conjectured that every perfect matching of a hypercube Qn for n≥2 can be extended to a hamiltonian cycle of Qn. Fink confirmed the conjecture to be true. It is more general to ask whether every perfect matching of Qn for n≥2 can be extended to two or more hamiltonian cycles of Qn. In this paper, we prove that every perfect matching of Qn for n≥4 can be extended to at least 22n−4 different hamiltonian cycles of Qn.

Hamiltonian cycles in hypercubes with faulty edges

ABSTRACTSzepietowski [12] observed that the hypercube is not Hamiltonian if it contains a trap disconnected halfway. A proper subgraph T is disconnected halfway if at least half of its nodes have parity 0 (or 1, resp.) and all edges joining the nodes of parity 0 (or 1, resp.) in T with nodes outside T, are faulty. In this paper, we describe all traps disconnected halfway T with the size and we consider the problem whether there exist small sets of faulty edges that are not based on sets disconnected halfway and still preclude Hamiltonian cycles. We show that if with the set of faulty edges F contains a trap disconnected halfway T that is of size and is minimal, then T is a path or is Hamiltonian. We also describe heuristic that recognizes nonhamiltonian cubes, also these ones that do not contain traps disconnected halfway.

Hamiltonian cycles in hypercubes with more faulty edges

ABSTRACTThe hypercube is one of the best known interconnection networks for parallel and distributed systems. In this paper we consider hamiltonian cycles in hypercubes with faulty edges. It is proved that there still exists a fault-free hamiltonian cycle in the hypercube with 3n−7 faulty edges if the following two conditions are satisfied: (1) the degree of every vertex is at least two, (2) there do not exist a forbidden -cycle or -cycle. Our result is optimal in some sense.

Small Matchings Extend to Hamiltonian Cycles in Hypercubes

Ruskey and Savage asked the following question: does every matching of a hypercube $$Q_{n}$$Qn for $$n\ge 2$$n?2 extend to a Hamiltonian cycle of $$Q_{n}$$Qn? Fink confirmed that the question is true for every perfect matching, thus solved Kreweras' conjecture. In this paper we prove that every matching of at most $$3n-10$$3n-10 edges can be extended to a Hamiltonian cycle of $$Q_{n}$$Qn for $$n\ge 4$$n?4.

Prescribed matchings extend to Hamiltonian cycles in hypercubes with faulty edges

Ruskey and Savage asked the following question: for n≥2, does every matching in Qn extend to a Hamiltonian cycle in Qn? Fink showed that the answer is yes for every perfect matching, thereby proving Kreweras’ conjecture. In this paper we consider the question in hypercubes with faulty edges. We show for n≥2 that every matching M of at most 2n−1 edges extends to a Hamiltonian cycle in Qn. Moreover, we prove that when n≥4 and M is nonempty this conclusion still holds even if Qn has at most n−1−⌈|M|2⌉ faulty edges, with one exception.

Hamiltonian cycles in hypercubes with faulty edges

Let F be a set of faulty edges in hypercube Qn with |F|⩽3n-8 for n⩾5. We prove that there still exists a fault-free Hamiltonian cycle in Qn if the following two conditions are satisfied: (1) the degree of every vertex is at least two, and (2) there do not exist a pair of nonadjacent vertices in a 4-cycle whose degrees are both two after faulty edges are removed.

Hamiltonian cycles in hypercubes with [formula omitted] faulty edges

In this paper we consider Hamiltonian cycles in hypercubes with faulty edges and present a simple proof that the hypercube Qn is not Hamiltonian if it contains a subgraph T such that: (i) exactly half of the vertices in T have parity 0 (the other half have parity 1) and (ii) all edges joining the nodes of parity 0 (or 1) with nodes outside T, are faulty. We will call such a subgraph a trap disconnected halfway. Moreover, we show that for n⩾5, the cube Qn with 2n-4 faulty edges is not Hamiltonian only if it contains a trap T disconnected halfway with T being a subcube of dimension 1 or 2.

A note on fault-free mutually independent Hamiltonian cycles in hypercubes with faulty edges

In the paper “Fault-free Mutually Independent Hamiltonian Cycles in Hypercubes with Faulty Edges” (J. Comb. Optim. 13:153–162, 2007), the authors claimed that an n-dimensional hypercube can be embedded with (n−1−f)-mutually independent Hamiltonian cycles when f≤n−2 faulty edges may occur accidentally. However, there are two mistakes in their proof. In this paper, we give examples to explain why the proof is deficient. Then we present a correct proof.

Fault-free mutually independent Hamiltonian cycles in hypercubes with faulty edges

Two Hamiltonian paths are said to be fully independent if the ith vertices of both paths are distinct for all i between 1 and n, where n is the number of vertices of the given graph. Hamiltonian paths in a set are said to be mutually fully independent if two arbitrary Hamiltonian paths in the set are fully independent. On the other hand, two Hamiltonian cycles are independent starting atv if both cycles start at a common vertex v and the ith vertices of both cycles are distinct for all i between 2 and n. Hamiltonian cycles in a set are said to be mutually independent starting atv if any two different cycles in the set are independent starting at v. The n-dimensional hypercube is widely used as the architecture for parallel machines. In this paper, we study its fault-tolerant property and show that an n-dimensional hypercube with at most n−2 faulty edges can embed a set of fault-free mutually fully independent Hamiltonian paths between two adjacent vertices, and can embed a set of fault-free mutually independent Hamiltonian cycles starting at a given vertex. The number of tolerable faulty edges is optimal with respect to a worst case.

Edge disjoint hamiltonian cycles in k-ary n-cubes and hypercubes

Solutions for decomposing a higher dimensional torus to edge disjoint lower dimensional tori, in particular, edge disjoint Hamiltonian cycles are obtained based on the coding theory approach. First, Lee distance Gray codes in Z/sub k//sup n/ are presented and then it is shown how these codes can directly be used to generate edge disjoint Hamiltonian cycles in k-ary n-cubes. Further, some new classes of binary Gray codes are designed from these Lee distance Gray codes and, using these new classes of binary Gray codes, edge disjoint Hamiltonian cycles in hypercubes are generated.

Classification of the Hamiltonian Cycles in Binary Hypercubes

In contrast to the conventional representation of the Hamiltonian cycles in the hypercubes as ring sequences of the numbers of the vertices, the paper proposed to represent the weights of edges connecting the pairs of adjacent cycle vertices as the ring sequences. The weight of an edge is the difference between the numbers of its incident vertices. Relying on the representation of Hamiltonian cycles by sequences of the edge weights, decomposition of the cycle set into classes defined by the distributions of numbers of different edge weights, as well as into kinds belonging to classes and defined by the distributions of edge weights was proposed. It was shown how by the operations of shift and permutation of edge weight one can obtain from the known sequence of edge weights representing some Hamiltonian cycle in the i>n-dimensional cube at least i>n!-1 other Hamiltonian cycles of the same class and kind. It was shown how one can pass from the class and kind of the Hamiltonian cycles for the i>n-dimensional cube to a “similar” class and kind of the Hamiltonian cycles for the (i>n+1)-dimensional cube.

EMBEDDING HAMILTONIAN CYCLES, LINEAR ARRAYS AND RINGS IN A FAULTY SUPERCUBE

We consider the problem of finding Hamiltonian cycles, linear arrays and rings of a faulty supercube, if any. The proof of the existence of Hamiltonian cycles in hypercubes is easy due to the fact they are symmetric graphs. Since the supercube is asymmetric, the proof of the existence of Hamiltonian cycles is not trivial. We show that for any supercube SN, where N is the number of nodes in the supercube, there exists a Hamiltonian cycle. This implies that for any r such that 3≤r≤N, there exists a cycle of r nodes in a supercube. There are embedding algorithms proposed in this paper. The embedding algorithms show a ring with any number of nodes which can be embedded in a faulty supercube with load 1, congestion 1 and dilation 4 such that O(n2-(⌊log2 m⌋)2) faults can be tolerated, where n is the dimension of the supercube and m is the number of nodes of the ring.