Cycles passing through a prescribed path in a hypercube with faulty edges

Let Qn denote an n-dimensional hypercube with n?2, P be a path of length h in Qn and F⊂E(Qn)\E(P). Recently, Tsai proved that if 1?h?n−1 and |F|?n−1−h, then in the graph Qn−F the path P lies on a cycle of every even length from 2h+2 to n², and P also lies on a cycle of length 2h if |F|?h−2. In this paper, we show that if 1?h?2n−3 and |F|?n−2−⌊h/2⌋, then in Qn−F the path P lies on a cycle of every even length from 2h+2 to n², and P also lies on a cycle of length 2h if P contains two edges of the same dimension or P is a shortest path and |F∩E(Qh)|?h−2, where Qh is the h-dimensional subcube containing the path P. Moreover, the upper bound 2n−3 of h is sharp and the upper bound n−2−⌊h/2⌋ of |F| is sharp for any given h with 1?h?2n−3.     

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.     

Edge-fault-tolerant diameter and bipanconnectivity of hypercubes

Let n(?3) be a given integer and . And let Qn be an n-dimensional hypercube and F⊂E(Qn), such that every vertex of the graph Qn−F is incident with at least two edges. Assume x and y are any two vertices with Hamming distance H(x,y)=h. In this paper, we obtain the following results: (1) If h?2 and |F|?min{n+h−1,2n−5}, then in Qn−F there exists an xy-path of each length l∈Ωh+2, and the upper bound n+h−1 on |F| is sharp when 2?h?n−4, and the upper bound 2n−5 on |F| is sharp when n−4?h?n−1 and h=2. (2) If |F|?2n−5, then in Qn−F there exists an xy-path of each length l∈Ωs, where s=h if n−1?h?n, and s=h+2 if n−4?h?n−2 and h?2, and s=h+4 otherwise. Hence, the diameter of the graph Qn−F is n. Our results improve some previous results.     

Maximum number of edges joining vertices on a cube

Let E_d(n) be the number of edges joining vertices from a set of n vertices on a d-dimensional cube, maximized over all such sets. We show that E_d(n)=∑_i=0^r−1(l_i/2+i)2^l_i, where r and l₀>l₁>?>l_r−1 are nonnegative integers defined by n=∑_i=0^r−12^l_i.     

Edge-fault-tolerant edge-bipancyclicity of hypercubes

In this paper, we consider the problem embedding a cycle into the hypercube Qn with existence of faulty edges and show that for any edge subset F of Qn with |F|?n−1 every edge of Qn−F lies on a cycle of every even length from 6 to n² inclusive provided n?4 and all edges in F are not incident with the same vertex. This result improves some known results.     

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.     

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.     

Bipanconnectivity and edge-fault-tolerant bipancyclicity of hypercubes

A bipartite graph is bipancyclic if it contains a cycle of every even length from 4 to |V(G)| inclusive. It has been shown that Q_n is bipancyclic if and only if n?2. In this paper, we improve this result by showing that every edge of Q_n−E′ lies on a cycle of every even length from 4 to |V(G)| inclusive where E′ is a subset of E(Q_n) with |E′|?n−2. The result is proved to be optimal. To get this result, we also prove that there exists a path of length l joining any two different vertices x and y of Q_n when h(x,y)?l?|V(G)|−1 and l−h(x,y) is even where h(x,y) is the Hamming distance between x and y.     

Edge-bipancyclicity of star graphs with faulty elements

In this paper, we investigate the fault-tolerant edge-bipancyclicity of an n-dimensional star graph S_n. Given a set F comprised of faulty vertices and/or edges in S_n with |F|≤n−3 and any fault-free edge e in S_n−F, we show that there exist cycles of every even length from 6 to n!−2|F_v| in S_n−F containing e, where n≥3. Our result is optimal because the star graph is both bipartite and regular with the common degree n−1. The length of the longest fault-free cycle in the bipartite S_n is n!−2|F_v| in the worst case, where all faulty vertices are in the same partite set. We also provide some sufficient conditions from which longer cycles with length from n!−2|F_v|+2 to n!−2|F_v| can be embedded.     

An efficient fault-tolerant routing algorithm in bijective connection networks with restricted faulty edges

In this paper, we study fault-tolerant routing in bijective connection networks with restricted faulty edges. First, we prove that the probability that all the incident edges of an arbitrary node become faulty in an n-dimensional bijective connection network, denoted by X_n, is extremely low when n becomes sufficient large. Then, we give an O(n) algorithm to find a fault-free path of length at most n+3⌈log₂∣F∣⌉+1 between any two different nodes in X_n if each node of X_n has at least one fault-free incident edge and the number of faulty edges is not more than 2n−3. In fact, we also for the first time provide an upper bound of the fault diameter of all the bijective connection networks with the restricted faulty edges. Since the family of BC networks contains hypercubes, crossed cubes, Möbius cubes, etc., all the results are appropriate for these cubes.     

Pancyclicity and bipancyclicity of conditional faulty folded hypercubes

A graph is said to be pancyclic if it contains cycles of every length from its girth to its order inclusive; and a bipartite graph is said to be bipancyclic if it contains cycles of every even length from its girth to its order. The pancyclicity or the bipancyclicity of a given network is an important factor in determining whether the network’s topology can simulate rings of various lengths. An n-dimensional folded hypercube FQ_n is an attractive variant of an n-dimensional hypercube Q_n that is obtained by establishing some extra edges between the vertices of Q_n. FQ_n for any odd n is known to be bipartite. In this paper, we explore the pancyclicity and bipancyclicity of FQ_n. For any FQ_n (n ? 2) with at most 2n − 3 faulty edges, where each vertex is incident to at least two fault-free edges, we prove that there exists a fault-free cycle of every even length from 4 to 2ⁿ; and when n ? 2 is even, we prove there also exists a fault-free cycle of every odd length from n + 1 to 2ⁿ − 1. The result is optimal with respect to the number of faulty edges tolerated.     

Fault-Tolerant Embedding of Pairwise Independent Hamiltonian Paths on a Faulty Hypercube with Edge Faults

A Hamiltonian path in G is a path which contains every vertex of G exactly once. Two Hamiltonian paths P ₁=〈u ₁,u ₂,…,u _n 〉 and P ₂=〈v ₁,v ₂,…,v _n 〉 of G are said to be independent if u ₁=v ₁, u _n =v _n , and u _i ≠v _i for all 1<i<n; and both are full-independent if u _i ≠v _i for all 1≤i≤n. Moreover, P ₁ and P ₂ are independent starting at u ₁, if u ₁=v ₁ and u _i ≠v _i for all 1<i≤n. A set of Hamiltonian paths {P ₁,P ₂,…,P _k } of G are pairwise independent (respectively, pairwise full-independent, pairwise independent starting at u ₁) if any two different Hamiltonian paths in the set are independent (respectively, full-independent, independent starting at u ₁). A bipartite graph G is Hamiltonian-laceable if there exists a Hamiltonian path between any two vertices from different partite sets. It is well known that an n-dimensional hypercube Q _n is bipartite with two partite sets of equal size. Let F be the set of faulty edges of Q _n . In this paper, we show the following results:

Hamiltonian paths and cycles passing through a prescribed path in hypercubes

Assume that P is any path in a bipartite graph G of length k with 2?k?h, G is said to be h-path bipancyclic if there exists a cycle C in G of every even length from 2k to |V(G)| such that P lies in C. In this paper, the following result is obtained: The n-dimensional hypercube Qn with n?3 is (2n−3)-path bipancyclic but is not (2n−2)-path bipancyclic, moreover, a path P of length k with 2?k?2n−3 lies in a cycle of length 2k−2 if and only if P contains two edges of the same dimension. In order to prove the above result we first show that any path of length at most 2n−1 is a subpath of a Hamiltonian path in Qn with n?2, moreover, the upper bound 2n−1 is sharp when n?4.     

A further result on fault-free cycles in faulty folded hypercubes

Folded hypercube is a well-known variation of the hypercube structure and can be constructed from a hypercube by adding a link to every pair of nodes with complementary addresses. Let FFv (respectively, FFe) be the set of faulty nodes (respectively, faulty links) in an n-dimensional folded hypercube FQn. Fu has showed that FQn−FFv−FFe for n?3 contains a fault-free cycle of length at least n²−2|FFv| if |FFv|+|FFe|?2n−4 and |FFe|?n−1. In this paper, we further consider the constraints |FFv|+|FFe|?2n−4 and |FFe|?n that were not covered by Fu's result. We obtain the same lower bound of the longest fault-free cycle length, n²−2|FFv|, under the constraints that (1) |FFv|+|FFe|?2n−4 and (2) every node in FQn is incident to at least two fault-free links.     

Edge-bipancyclicity of the k-ary n-cubes with faulty nodes and edges

The k-ary n-cube has been one of the most popular interconnection networks for massively parallel systems. In this paper, we investigate the edge-bipancyclicity of k-ary n-cubes with faulty nodes and edges. It is proved that every healthy edge of the faulty k-ary n-cube with f_v faulty nodes and f_e faulty edges lies in a fault-free cycle of every even length from 4 to kⁿ − 2f_v (resp. kⁿ − f_v) if k ? 4 is even (resp. k ? 3 is odd) and f_v + f_e ? 2n − 3. The results are optimal with respect to the number of node and edge faults tolerated.     

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.     

Ring embedding in faulty pancake graphs

In this paper, we consider the fault hamiltonicity and the fault hamiltonian connectivity of the pancake graph P_n. Assume that F⊆V(P_n)∪E(P_n). For n?4, we prove that P_n−F is hamiltonian if |F|?(n−3) and P_n−F is hamiltonian connected if |F|?(n−4). Moreover, all the bounds are optimal.     

Geometric pattern matching for point sets in the plane under similarity transformations

We consider the following geometric pattern matching problem: Given two sets of points in the plane, P and Q, and some (arbitrary) δ>0, find a similarity transformation T (translation, rotation and scale) such that h(T(P),Q)<δ, where h(⋅,⋅) is the directional Hausdorff distance with L_∞ as the underlying metric; or report that none exists. We are only interested in the decision problem, not in minimizing the Hausdorff distance, since in the real world, where our applications come from, δ is determined by the practical uncertainty in the position of the points (pixels). Similarity transformations have not been dealt with in the context of the Hausdorff distance and we fill the gap here. We present efficient algorithms for this problem imposing a reasonable separation restriction on the points in the set Q. If the L_∞ distance between every pair of points in Q is at least 8δ, then the problem can be solved in O(mn²logn) time, where m and n are the numbers of points in P and Q respectively. If the L_∞ distance between every pair of points in Q is at least cδ, for some c, 0<c<1, we present a randomized approximate solution with expected runtime O(n²c−4ε−8log⁴mn), where ε>0 controls the approximation. Our approximation is on the size of the subset, B⊆P, such that h(T(B),Q)<δ and |B|>(1−ε)|P| with high probability.     

Embedding two edge-disjoint Hamiltonian cycles into locally twisted cubes

The n-dimensional hypercube network Q_n is one of the most popular interconnection networks since it has simple structure and is easy to implement. The n-dimensional locally twisted cube LTQ_n, an important variation of the hypercube, has the same number of nodes and the same number of connections per node as Q_n. One advantage of LTQ_n is that the diameter is only about half of the diameter of Q_n. Recently, some interesting properties of LTQ_n have been investigated in the literature. The presence of edge-disjoint Hamiltonian cycles provides an advantage when implementing algorithms that require a ring structure by allowing message traffic to be spread evenly across the interconnection network. The existence of two edge-disjoint Hamiltonian cycles in locally twisted cubes has remained unknown. In this paper, we prove that the locally twisted cube LTQ_n with n?4 contains two edge-disjoint Hamiltonian cycles. Based on the proof of existence, we further provide an O(n2ⁿ)-linear time algorithm to construct two edge-disjoint Hamiltonian cycles in an n-dimensional locally twisted cube LTQ_n with n?4, where LTQ_n contains 2ⁿ nodes and n2ⁿ⁻¹ edges.     

Embedding meshes into locally twisted cubes

As a newly introduced interconnection network for parallel computing, the locally twisted cube possesses many desirable properties. In this paper, mesh embeddings in locally twisted cubes are studied. Let LTQ_n(V, E) denote the n-dimensional locally twisted cube. We present three major results in this paper: (1) For any integer n ? 1, a 2 × 2ⁿ⁻¹ mesh can be embedded in LTQ_n with dilation 1 and expansion 1. (2) For any integer n ? 4, two node-disjoint 4 × 2ⁿ⁻³ meshes can be embedded in LTQ_n with dilation 1 and expansion 2. (3) For any integer n ? 3, a 4  × (2ⁿ⁻² − 1) mesh can be embedded in LTQ_n with dilation 2. The first two results are optimal in the sense that the dilations of all embeddings are 1. The embedding of the 2 × 2ⁿ⁻¹ mesh is also optimal in terms of expansion. We also present the analysis of 2p × 2q mesh embedding in locally twisted cubes.