Hamiltonian paths and cycles with prescribed edges in the 3-ary n-cube

The k-ary n-cube has been one of the most popular interconnection networks for massively parallel systems. Given a set P of at most 2n − 2 (n ? 2) prescribed edges and two vertices u and v, we show that the 3-ary n-cube contains a Hamiltonian path between u and v passing through all edges of P if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths, none of them having u or v as internal vertices or both of them as end-vertices. As an immediate result, the 3-ary n-cube contains a Hamiltonian cycle passing through a set P of at most 2n − 1 prescribed edges if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths.     

On edge connectivity of direct products of graphs

Let λ(G) be the edge connectivity of G. The direct product of graphs G and H is the graph with vertex set V(G×H)=V(G)×V(H), where two vertices (u₁,v₁) and (u₂,v₂) are adjacent in G×H if u₁u₂∈E(G) and v₁v₂∈E(H). We prove that λ(G×Kn)=min{n(n−1)λ(G),(n−1)δ(G)} for every nontrivial graph G and n?3. We also prove that for almost every pair of graphs G and H with n vertices and edge probability p, G×H is k-connected, where k=O(²(n/logn)).     

Paths in Möbius cubes and crossed cubes

The Möbius cube MQn and the crossed cube CQn are two important variants of the hypercube Qn. This paper shows that for any two different vertices u and v in G∈{MQn,CQn} with n?3, there exists a uv-path of every length from dG(u,v)+2 to n²−1 except for a shortest uv-path, where dG(u,v) is the distance between u and v in G. This result improves some known results.     

Cycles embedding in hypercubes with node failures

The hypercube has been widely used as the interconnection network in parallel computers. The n-dimensional hypercube Qn is a graph having n² vertices each labeled with a distinct n-bit binary strings. Two vertices are linked by an edge if and only if their addresses differ exactly in the one bit position. Let fv denote the number of faulty vertices in Qn. For n?3, in this paper, we prove that every fault-free edge and fault-free vertex of Qn lies on a fault-free cycle of every even length from 4 to n²−2fv inclusive even if fv?n−2. Our results are optimal.     

Hamiltonian laceability of bubble-sort graphs with edge faults

It is known that the n-dimensional bubble-sort graph B_n is bipartite, (n − 1)-regular, and has n! vertices. We first show that, for any vertex v, B_n − v has a hamiltonian path between any two vertices in the same partite set without v. Let F be a subset of edges of B_n. We next show that B_n − F has a hamiltonian path between any two vertices of different partite sets if ∣F∣ is at most n − 3. Then we also prove that B_n − F has a path of length n! − 2 between any pair of vertices in the same partite set.     

Hamiltonian connectivity and globally 3∗-connectivity of dual-cube extensive networks

In 2000, Li et al. introduced dual-cube networks, denoted by DC_n for n?1, using the hypercube family Q_n and showed the vertex symmetry and some fault-tolerant hamiltonian properties of DC_n. In this article, we introduce a new family of interconnection networks called dual-cube extensive networks, denoted by DCEN(G). Given any arbitrary graph G, DCEN(G) is generated from G using the similar structure of DC_n. We show that if G is a nonbipartite and hamiltonian connected graph, then DCEN(G) is hamiltonian connected. In addition, if G has the property that for any two distinct vertices u,v of G, there exist three disjoint paths between u and v such that these three paths span the graph G, then DCEN(G) preserves the same property. Furthermore, we prove that the similar results hold when G is a bipartite graph.     

The edge-orientation problem and some of its variants on weighted graphs

Let G(V, E) be a connected undirected graph with n vertices and m edges, where each vertex v is associated with a cost C(v) and each edge e = (u, v) is associated with two weights, W(u → v) and W(v → u). The issue of assigning an orientation to each edge so that G becomes a directed graph is resolved in this paper. Determining a scheme to assign orientations of all edges such that max_x∈V{C(x)+∑_x→zW(x→z)} is minimized is the objective. This issue is called the edge-orientation problem (the EOP). Two variants of the EOP, the Out-Degree-EOP and the Vertex-Weighted EOP, are first proposed and then efficient algorithms for solving them on general graphs are designed. Ascertaining that the EOP is NP-hard on bipartite graphs and chordal graphs is the second result. Finally, an O(n log n)-time algorithm for the EOP on trees is designed. In general, the algorithmic results in this paper facilitate the implementation of the weighted fair queuing (WFQ) on real networks. The objective of the WFQ is to assign an effective weight for each flow to enhance link utilization. Our findings consequently can be easily extended to other classes of graphs, such as cactus graphs, block graphs, and interval graphs.     

Fault-tolerant hamiltonian laceability of hypercubes

It is known that every hypercube Q_n is a bipartite graph. Assume that n?2 and F is a subset of edges with |F|?n−2. We prove that there exists a hamiltonian path in Q_n−F between any two vertices of different partite sets. Moreover, there exists a path of length 2ⁿ−2 between any two vertices of the same partite set. Assume that n?3 and F is a subset of edges with |F|?n−3. We prove that there exists a hamiltonian path in Q_n−{v}−F between any two vertices in the partite set without v. Furthermore, all bounds are tight.     

Graph Decomposition for Memoryless Periodic Exploration

We consider a general framework in which a memoryless robot periodically explores all the nodes of a connected anonymous graph by following local information available at each vertex. For each vertex?v, the endpoints of all edges adjacent to v are assigned unique labels within the range?1 to?deg?(v) (the degree of?v). The generic exploration strategy is implemented using a right-hand-rule transition function: after entering vertex v via the edge labeled?i, the robot proceeds with its exploration, leaving via the edge having label [i mod deg?(v)]+1 at v. A lot of attention has been given to the problem of labeling the graph so as to achieve a periodic exploration having the minimum possible length?π. It has recently been proved (Czyzowicz et?al., Proc.?SIROCCO’09, 2009) that $\pi\leq4\frac{1}{3}n$ holds for all graphs of n vertices. Herein, we provide a new labeling scheme which leads to shorter exploration cycles, improving the general bound to π≤4n?2. This main result is shown to be tight with respect to the class of labellings admitting certain connectivity properties. The labeling scheme is based on a new graph decomposition which may be of independent interest.     

Fault-tolerant analysis of a class of networks

In this paper, we explore the 2-extraconnectivity of a special class of graphs G(G₀,G₁;M) proposed by Chen et al. [Y.-C. Chen, J.J.M. Tan, L.-H. Hsu, S.-S. Kao, Super-connectivity and super edge-connectivity for some interconnection networks, Applied Mathematics and Computation 140 (2003) 245-254]. As applications of the results, we obtain that the 2-extraconnectivities of several well-known interconnection networks, such as hypercubes, twisted cubes, crossed cubes, Möbius cubes and locally twisted cubes, are all equal to 3n−5 when their dimension n is not less than 8. That is, when n?8, at least 3n−5 vertices must be removed to disconnect any one of these n-dimensional networks provided that the removal of these vertices does not isolate a vertex or an edge.     

Constructing the nearly shortest path in crossed cubes

A graph G is panconnected if each pair of distinct vertices u,v∈V(G) are joined by a path of length l for all d_G(u,v)?l?|V(G)|-1, where d_G(u,v) is the length of a shortest path joining u and v in G. Recently, Fan et. al. [J. Fan, X. Lin, X. Jia, Optimal path embedding in crossed cubes, IEEE Trans. Parall. Distrib. Syst. 16 (2) (2005) 1190-1200, J. Fan, X. Jia, X. Lin, Complete path embeddings in crossed cubes, Inform. Sci. 176 (22) (2006) 3332-3346] and Xu et. al. [J.M. Xu, M.J. Ma, M. Lu, Paths in Möbius cubes and crossed cubes, Inform. Proc. Lett. 97 (3) (2006) 94-97] both proved that n-dimensional crossed cube, CQ_n, is almost panconnected except the path of length d_CQn(u,v)+1 for any two distinct vertices u,v∈V(CQ_n). In this paper, we give a necessary and sufficient condition to check for the existence of paths of length d_CQn(u,v)+1, called the nearly shortest paths, for any two distinct vertices u,v in CQ_n. Moreover, we observe that only some pair of vertices have no nearly shortest path and we give a construction scheme for the nearly shortest path if it exists.     

An efficient distributed algorithm for finding all hinge vertices in networks

Let G=(V, E) be a graph with vertex set V of size n and edge set E of size m. A vertex v∈V is called a hinge vertex if there exist two vertices in V\{v} such that their distance becomes longer when v is removed. In this paper, we present a distributed algorithm that finds all hinge vertices on an arbitrary graph. The proposed algorithm works for named static asynchronous networks and achieves O(n ²) time complexity and O(m) message complexity. In particular, the total messages exchanged during the algorithm are at most 2m(log n+nlog n+1) bits.     

Broadcast in the rendezvous model

In many large, distributed or mobile networks, broadcast algorithms are used to update information stored at the nodes. In this paper, we propose a new model of communication based on rendezvous and analyze a multi-hop distributed algorithm to broadcast a message in a synchronous setting. In the rendezvous model, two neighbors u and v can communicate if and only if u calls v and v calls u simultaneously. Thus nodes u and v obtain a rendezvous at a meeting point. If m is the number of meeting points, the network can be modeled by a graph of n vertices and m edges. At each round, every vertex chooses a random neighbor and there is a rendezvous if an edge has been chosen by its two extremities. Rendezvous enable an exchange of information between the two entities. We get sharp lower and upper bounds on the time complexity in terms of number of rounds to broadcast: we show that, for any graph, the expected number of rounds is between ln n and O (n²). For these two bounds, we prove that there exist some graphs for which the expected number of rounds is either O (ln (n)) or Ω (n²). For specific topologies, additional bounds are given.     

The super laceability of the hypercubes

A k-containerC(u,v) of a graph G is a set of k disjoint paths joining u to v. A k-container C(u,v) is a k∗-container if every vertex of G is incident with a path in C(u,v). A bipartite graph G is k∗-laceable if there exists a k∗-container between any two vertices u, v from different partite set of G. A bipartite graph G with connectivity k is super laceable if it is i∗-laceable for all i?k. A bipartite graph G with connectivity k is f-edge fault-tolerant super laceable if G−F is i∗-laceable for any 1?i?k−f and for any edge subset F with |F|=f<k−1. In this paper, we prove that the hypercube graph Q_r is super laceable. Moreover, Q_r is f-edge fault-tolerant super laceable for any f?r−2.     

Hamiltonian properties of twisted hypercube-like networks with more faulty elements

Twisted hypercube-like networks (THLNs) are a large class of network topologies, which subsume some well-known hypercube variants. This paper is concerned with the longest cycle in an n-dimensional (n-D) THLN with up to 2n−9 faulty elements. Let G be an n-D THLN, n≥7. Let F be a subset of V(G)?E(G), |F|≤2n−9. We prove that G−F contains a Hamiltonian cycle if δ(G−F)≥2, and G−F contains a near Hamiltonian cycle if δ(G−F)≤1. Our work extends some previously known results.     

All-Pairs Bottleneck Paths in Vertex Weighted Graphs

Let G=(V,E,w) be a directed graph, where w:V→ℝ is a weight function defined on its vertices. The bottleneck weight, or the capacity, of a path is the smallest weight of a vertex on the path. For two vertices u,v the capacity from u to v, denoted by c(u,v), is the maximum bottleneck weight of a path from u to v. In the All-Pairs Bottleneck Paths (APBP) problem the task is to find the capacities for all ordered pairs of vertices. Our main result is an O(n ^2.575) time algorithm for APBP. The exponent is derived from the exponent of fast matrix multiplication.     

A Fan-type result on k-ordered graphs

For a positive integer k, a graph G is k-ordered hamiltonian if for every ordered sequence of k vertices there is a hamiltonian cycle that encounters the vertices of the sequence in the given order. In this paper, we show that if G is a ⌊3k/2⌋-connected graph of order n?100k, and d(u)+d(v)?n for any two vertices u and v with d(u,v)=2, then G is k-ordered hamiltonian. Our result implies the theorem of G. Chen et al. [Ars Combin. 70 (2004) 245-255] [1], which requires the degree sum condition for all pairs of non-adjacent vertices, not just those distance 2 apart.     

A Quadratic Algorithm for Finding Next-to-Shortest Paths in Graphs

Given an edge-weighted undirected graph G and two prescribed vertices u and v, a next-to-shortest (u,v)-path is a shortest (u,v)-path amongst all (u,v)-paths having length strictly greater than the length of a shortest (u,v)-path. In this paper, we deal with the problem of computing a next-to-shortest (u,v)-path. We propose an O(n²){\mathcal{O}}(n^{2}) time algorithm for solving this problem, which significantly improves the bound of a previous one in O(n³){\mathcal{O}}(n^{3}) time where n is the number of vertices in G.     

On the k-tuple domination of de Bruijn and Kautz digraphs

In a digraph G, a vertex u is said to dominate itself and vertices v such that (u,v) is an arc of G. For a positive integer k, a k-tuple dominating set D of a digraph is a subset of vertices such that every vertex is dominated by at least k vertices in D. The k-tuple domination number of a given digraph is the minimum cardinality of a k-tuple dominating set of the digraph. In this letter, we give the exact values of the k-tuple domination number of de Bruijn and Kautz digraphs.     

On the k-tuple domination of generalized de Brujin and Kautz digraphs

A vertex u in a digraph G = (V, A) is said to dominate itself and vertices v such that (u, v) ∈ A. For a positive integer k, a k-tuple dominating set of G is a subset D of vertices such that every vertex in G is dominated by at least k vertices in D. The k-tuple domination number of G is the minimum cardinality of a k-tuple dominating set of G. This paper deals with the k-tuple domination problem on generalized de Bruijn and Kautz digraphs. We establish bounds on the k-tuple domination number for the generalized de Bruijn and Kautz digraphs and we obtain some conditions for the k-tuple domination number attaining the bounds.