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 {{\displaystyle Q}}_n, and obtain the following results. Let n\ge \mathrm{3},\hspace{0.17em}M\subset E({{\displaystyle Q}}_n), and F\subset E({{\displaystyle Q}}_n)\backslash M with \mathrm{1}\le \left|F\right|\le \mathrm{2}n-\mathrm{4}-\left|M\right|. If M is a matching and every vertex is incident with at least two edges in the graph {{\displaystyle Q}}_n-F, then all edges of M lie on a Hamiltonian cycle in {{\displaystyle Q}}_n-F. Moreover, if \left|M\right|=\mathrm{1} or \left|M\right|=\mathrm{2}, then the upper bound of number of faulty edges tolerated is sharp. Our results generalize the well-known result for \left|M\right|=\mathrm{1}.     

Perfect matchings extending on subcubes to Hamiltonian cycles of hypercubes

Recently, Fink [J. Fink, Perfect matchings extend to Hamilton cycles in hypercubes, J. Combin. Theory Ser. B 97 (2007) 1074-1076] affirmatively answered Kreweras’ conjecture asserting that every perfect matching of the hypercube extends to a Hamiltonian cycle. We strengthen this result in the following way. Given a partition of the hypercube into subcubes of nonzero dimensions, we show for every perfect matching of the hypercube that it extends on these subcubes to a Hamiltonian cycle if and only if it interconnects them.     

Perfect matchings extend to two or more hamiltonian cycles in hypercubes

Kreweras conjectured that every perfect matching of a hypercube $Q_n$ for $n\ge 2$ can be extended to a hamiltonian cycle of $Q_n$. Fink confirmed the conjecture to be true. It is more general to ask whether every perfect matching of $Q_n$ for $n\ge 2$ can be extended to two or more hamiltonian cycles of $Q_n$. In this paper, we prove that every perfect matching of $Q_n$ for $n\ge 4$ can be extended to at least $2^{2^{n?4}}$ different hamiltonian cycles of $Q_n$.     

The carvingwidth of hypercubes

The notion of the carvingwidth of a graph was introduced by Seymour and Thomas [Call routing and the ratcatcher, Combinatorica 14 (1994) 217-241]. In this note, we show that the carvingwidth of a d-dimensional hypercube equals 2^d-1.     

Cycles embedding on folded hypercubes with faulty nodes

Let F_Fv$F{F}_v$ be the set of faulty nodes in an n$n$-dimensional folded hypercube F_Qn$F{Q}_n$ with |F_Fv|≤n−2$\left|F{F}_v\right|\le n-2$. In this paper, we show that if n≥3$n\ge 3$, then every edge of F_Qn−F_Fv$F{Q}_n-F{F}_v$ lies on a fault-free cycle of every even length from 4$4$ to 2ⁿ−2|F_Fv|$2^n-2\left|F{F}_v\right|$, and if n≥2$n\ge 2$ and n$n$ is even, then every edge of F_Qn−F_Fv$F{Q}_n-F{F}_v$ lies on a fault-free cycle of every odd length from n+1$n+1$ to 2ⁿ−2|F_Fv|−1$2^n-2\left|F{F}_v\right|-1$.     

The treewidth and pathwidth of hypercubes

The d-dimensional hypercube, H_d, is the graph on 2^d vertices, which correspond to the 2^dd-vectors whose components are either 0 or 1, two of the vertices being adjacent when they differ in just one coordinate. The notion of Hamming graphs (denoted by ) generalizes the notion of hypercubes: The vertices correspond to the q^dd-vectors where the components are from the set {0,1,2,…,q-1}, and two of the vertices are adjacent if and only if the corresponding vectors differ in exactly one component. In this paper we show that the and the .     

On explicit formulas for bandwidth and antibandwidth of hypercubes

The Hales numbered n-dimensional hypercube exhibits interesting recursive structures in n. These structures lead to a very simple proof of the well-known bandwidth formula for hypercubes proposed by Harper, whose proof was thought to be surprisingly difficult. Harper also proposed an optimal numbering for a related problem called the antibandwidth of hypercubes. In a recent publication, Raspaud et al. approximated the hypercube antibandwidth up to the third-order term. In this paper, we find the exact value in light of the above recursive structures.     

Signless Laplacian spectral radius and Hamiltonicity of graphs with large minimum degree

In this paper, we establish a tight sufficient condition for the Hamiltonicity of graphs with large minimum degree in terms of the signless Laplacian spectral radius and characterize all extremal graphs. Moreover, we prove a similar result for balanced bipartite graphs. Additionally, we construct infinitely many graphs to show that results proved in this paper give new strength for one to determine the Hamiltonicity of graphs.     

The carving-width of generalized hypercubes

The carving-width of a graph is the minimum congestion of routing trees for the graph. We determine the carving-width of generalized hypercubes: Hamming graphs, even grids, and tori. Our results extend the result of Chandran and Kavitha [L.S. Chandran, T. Kavitha, The carvingwidth of hypercubes, Discrete Math. 306 (2006) 2270-2274] that determines the carving-width of hypercubes.     

Geodesic-pancyclic graphs

A shortest path connecting two vertices u and v is called a u-v geodesic. The distance between u and v in a graph G, denoted by d_G(u,v), is the number of edges in a u-v geodesic. A graph G with n vertices is panconnected if, for each pair of vertices u,v∈V(G) and for each integer k with d_G(u,v)?k?n-1, there is a path of length k in G that connects u and v. A graph G with n vertices is geodesic-pancyclic if, for each pair of vertices u,v∈V(G), every u-v geodesic lies on every cycle of length k satisfying max{2d_G(u,v),3}?k?n. In this paper, we study sufficient conditions of geodesic-pancyclic graphs. In particular, we show that most of the known sufficient conditions of panconnected graphs can be applied to geodesic-pancyclic graphs.     

Pancyclicity mod k of claw-free graphs and K1,4-free graphs

For any natural number k, a graph G is said to be pancyclic mod k if it contains a cycle of every length modulo k. In this paper, we show that every K_1,4-free graph G with minimum degree δ(G)k+3 is pancyclic mod k and every claw-free graph G with δ(G)k+1 is pancyclic mod k, which confirms Thomassen's conjecture (J. Graph Theory 7 (1983) 261–271) for claw-free graphs.     

On the multisearching problem for hypercubes

In this paper we give improved bounds for the multisearch problem on a hypercube. This is a parallel search problem where the elements in the structure S to be searched are totally ordered, but where it is not possible to compare in constant time any two given queries q and q′. More precisely, we are given on a n-processor hypercube a sorted n-element sequence S, and a set Q of n queries, and we need to find for each query q Q its location in the sorted S. We present an improved algorithm for the multisearch problem, one that takes O(log n(log log n)³) time on a n-processor hypercube. This problem is fundamental in computational geometry, for example it models planar point location in a slab. We give as application a trapezoidal decomposition algorithm with the same time complexity on a n log n-processor hypercube. The hypercube model for which we claim our bounds is the standard one, SIMD, with O(1) memory registers per processor, and with one-port communication. Each register can store O(log n) bits, so that a processor knows its ID.     

Recursive fault-tolerance of Fibonacci cube in hypercubes

Fibonacci cube is a subgraph of hypercube induced on vertices without two consecutive 1's. If we remove from Fibonacci cube the vertices with 1 both in the first and the last position, we obtain Lucas cube. We consider the problem of determining the minimum number of vertices in n-dimensional hypercube whose removal leaves no subgraph isomorphic to m-dimensional Fibonacci cube. The exact values for small m are given and several recursive bounds are established using the symmetry property of Lucas cubes and the technique of labeling. The relation to the problem of subcube fault-tolerance in hypercube is also shown.     

Edge-bipancyclicity of a hypercube with faulty vertices and edges

A bipartite graph G=(V,E) is said to be bipancyclic if it contains a cycle of every even length from 4 to |V|. Furthermore, a bipancyclic G is said to be edge-bipancyclic if every edge of G lies on a cycle of every even length. Let F_v (respectively, F_e) be the set of faulty vertices (respectively, faulty edges) in an n-dimensional hypercube Q_n. In this paper, we show that every edge of Q_n-F_v-F_e lies on a cycle of every even length from 4 to 2ⁿ-2|F_v| even if |F_v|+|F_e|?n-2, where n?3. Since Q_n is bipartite of equal-size partite sets and is regular of vertex-degree n, both the number of faults tolerated and the length of a longest fault-free cycle obtained are worst-case optimal.     

On the size of identifying codes in binary hypercubes

In this paper, we consider identifying codes in binary Hamming spaces Fn, i.e., in binary hypercubes. The concept of (r,??)-identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. Currently, the subject forms a topic of its own with several possible applications, for example, to sensor networks.Let us denote by the smallest possible cardinality of an (r,??)-identifying code in Fn. In 2002, Honkala and Lobstein showed for ?=1 that

     

Antibandwidth and cyclic antibandwidth of meshes and hypercubes

The antibandwidth problem consists of placing the vertices of a graph on a line in consecutive integer points in such a way that the minimum difference of adjacent vertices is maximised. The problem was originally introduced in [J.Y.-T. Leung, O. Vornberger, J.D. Witthoff, On some variants of the bandwidth minimisation problem, SIAM Journal of Computing 13 (1984) 650-667] in connection with the multiprocessor scheduling problems and can also be understood as a dual problem to the well-known bandwidth problem, as a special radiocolouring problem or as a variant of obnoxious facility location problems. The antibandwidth problem is NP-hard, there are a few classes of graphs with polynomial time complexities. Exact results for nontrivial graphs are very rare. Miller and Pritikin [Z. Miller, D. Pritikin, On the separation number of a graph, Networks 19 (1989) 651-666] showed tight bounds for the two-dimensional meshes and hypercubes. We solve the antibandwidth problem precisely for two-dimensional meshes, tori and estimate the antibandwidth value for hypercubes up to the third-order term. The cyclic antibandwidth problem is to embed an n-vertex graph into the cycle C_n, such that the minimum distance (measured in the cycle) of adjacent vertices is maximised. This is a natural extension of the antibandwidth problem or a dual problem to the cyclic bandwidth problem. We start investigating this invariant for typical graphs and prove basic facts and exact results for the same product graphs as for the antibandwidth.     

A new upper bound on the queuenumber of hypercubes

A queue layout of a graph consists of a linear order of its vertices, and a partition of its edges into queues, such that no two edges in the same queue are nested. In this paper, we show that the n-dimensional hypercube Q_n can be laid out using n−3 queues for n?8. Our result improves the previously known result for the case n?8.