Paired many-to-many disjoint path covers in faulty hypercubes (original) (raw)
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Information Sciences, 2013
A paired many-to-many k-disjoint path cover (paired k-DPC for short) of a graph is a set of k vertex-disjoint paths joining k distinct source-sink pairs that altogether cover every vertex of the graph. We consider the problem of constructing paired 2-DPC's in an m-dimensional bipartite HL-graph, X m , and its application in finding the longest possible paths. It is proved that every X m , m ≥ 4, has a fault-free paired 2-DPC if there are at most m − 3 faulty edges and the set of sources and sinks is balanced in the sense that it contains the same number of vertices from each part of the bipartition. Furthermore, every X m , m ≥ 4, has a paired 2-DPC in which the two paths have the same length if each source-sink pair is balanced. Using 2-DPC properties, we show that every X m , m ≥ 3, with either at most m − 2 faulty edges or one faulty vertex and at most m − 3 faulty edges is strongly Hamiltonian-laceable.
Many-to-many two-disjoint path covers in restricted hypercube-like graphs
Theoretical Computer Science, 2014
A Disjoint Path Cover (DPC for short) of a graph is a set of pairwise (internally) disjoint paths that altogether cover every vertex of the graph. Given a set S of k sources and a set T of k sinks, a many-to-many k-DPC between S and T is a disjoint path cover each of whose paths joins a pair of source and sink. It is classified as paired if each source of S must be joined to a designated sink of T , or unpaired if there is no such constraint. In this paper, we show that every m-dimensional restricted hypercube-like graph with at most m − 3 faulty vertices and/or edges being removed has a paired (and unpaired) 2-DPC joining arbitrary two sources and two sinks where m ≥ 5. The bound m − 3 on the number of faults is optimal for both paired and unpaired types.
Paired many-to-many disjoint path covers in restricted hypercube-like graphs
Theoretical Computer Science, 2016
Given two disjoint vertex-sets, S = {s 1 ,. .. , s k } and T = {t 1 ,. .. , t k } in a graph, a paired many-to-many k-disjoint path cover between S and T is a set of pairwise vertex-disjoint paths {P 1 ,. .. , P k } that altogether cover every vertex of the graph, in which each path P i runs from s i to t i. A family of hypercube-like interconnection networks, called restricted hypercube-like graphs, includes most non-bipartite hypercube-like networks found in the literature, such as twisted cubes, crossed cubes, Möbius cubes, recursive circulant G(2 m , 4) of odd m, etc. In this paper, we show that every m-dimensional restricted hypercube-like graph, m ≥ 5, with at most f vertex and/or edge faults being removed has a paired many-to-many k-disjoint path cover between arbitrary disjoint sets S and T of size k each, subject to k ≥ 2 and f + 2k ≤ m + 1. The bound m + 1 on f + 2k is the best possible.
Unpaired many-to-many disjoint path covers in restricted hypercube-like graphs
Theoretical Computer Science, 2016
For two disjoint vertex-sets, S = {s 1 ,. .. , s k } and T = {t 1 ,. .. , t k } of a graph, an unpaired many-to-many k-disjoint path cover joining S and T is a set of pairwise vertex-disjoint paths {P 1 ,. .. , P k } that altogether cover every vertex of the graph, in which P i is a path from s i to some t j for 1 ≤ i, j ≤ k. A family of hypercube-like interconnection networks, called restricted hypercube-like graphs, includes most nonbipartite hypercube-like networks found in the literature, such as twisted cubes, crossed cubes, Möbius cubes, recursive circulant G(2 m , 4) of odd m, etc. In this paper, we show that every m-dimensional restricted hypercube-like graph, m ≥ 5, with at most f faulty vertices and/or edges being removed has an unpaired many-to-many k-disjoint path cover joining arbitrary disjoint sets S and T of size k each subject to k ≥ 2 and f + k ≤ m − 1. The bound m − 1 on f + k is the maximum possible.
Hamiltonicity of hypercubes with a constraint of required and faulty edges
Journal of Combinatorial Optimization, 2007
Let R and F be two disjoint edge sets in an n-dimensional hypercube Q n . We give two constructing methods to build a Hamiltonian cycle or path that includes all the edges of R but excludes all of F . Besides, considering every vertex of Q n incident to at most n − 2 edges of F , we show that a Hamiltonian cycle exists if (A) |R| + 2|F | ≤ 2n − 3 when |R| ≥ 2, or (B) |R| + 2|F | ≤ 4n − 9 when |R| ≤ 1. Both bounds are tight. The analogous property for Hamiltonian paths is also given.
Hamiltonian cycles and paths in hypercubes with disjoint faulty edges
Inf. Process. Lett., 2021
We consider hypercubes with pairwise disjoint faulty edges. An nnn-dimensional hypercube QnQ_nQn is an undirected graph with 2n2^n2n nodes, each labeled with a distinct binary strings of length nnn. The parity of the vertex is 0 if the number of ones in its labels is even, and is 1 if the number of ones is odd. Two vertices aaa and bbb are connected by the edge iff aaa and bbb differ in one position. If aaa and bbb differ in position iii, then we say that the edge (a,b)(a,b)(a,b) goes in direction iii and we define the parity of the edge as the parity of the end with 0 on the position iii. It was already known that QnQ_nQn is not Hamiltonian if all edges going in one direction and of the same parity are faulty. In this paper we show that if nge4n\ge4nge4 then all other hypercubes are Hamiltonian. In other words, every cube QnQ_nQn, with nge4n\ge4nge4 and disjoint faulty edges is Hamiltonian if and only if for each direction there are two healthy crossing edges of different parity.
On the Existence of Disjoint Spanning Paths in Faulty Hypercubes
Journal of Interconnection Networks, 2010
Assume that n is a positive integer with n ≥ 4 and F is a subset of the edges of the hypercube Qn with |F| ≤ n-4. Let u , x be two distinct white vertices of Qn and v , y be two distinct black vertices of Qn, where black and white refer to the two parts of the bipartition of Qn. Let l1 and l2 be odd integers, where l1 ≥ dQn-F( u , v ), l2 ≥ dQn-F( x , y ), and l1 + l2 = 2n - 2. Moreover, let l3 and l4 be even integers, where l3 ≥ dQn-F( u , x ), l4 ≥ dQn-F( v , y ), and l3+l4 = 2n - 2. In this paper, we prove that there are two disjoint paths P1 and P2 such that (1) P1 is a path joining u to v with length l(P1) = l1, (2) P2 is a path joining x to y with l(P2) = l2, and (3) P1 ∪ P2 spans Qn - F. Moreover, there are two disjoint paths P3 and P4 such that (1) P3 is a path joining u to x with l(P3) = l3, (2) P4 is a path joining v to y with l(P4) = l4, and (3) P3 ∪ P4 spans Qn - F except the following cases: (a) l3 = 2 with dQn-F( u , x ) = 2 and dQn-F-{ v , y }( u , x ) > 2, and (b)...
Hamiltonian paths and cycles pass through prescribed edges in the balanced hypercubes
Discrete Applied Mathematics, 2019
The n-dimensional balanced hypercube BH n (n ≥ 1) has been proved to be a bipartite graph. Let P be a set of edges in BH n. For any two vertices u, v from different partite sets of V (BH n). In this paper, we prove that if |P| ≤ 2n − 2 and the subgraph induced by P has neither u nor v as internal vertices , or both of u and v as end-vertices, then BH n contains a Hamiltonian path joining u and v passing through every edge of P if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths. As a corollary, if |P| ≤ 2n − 1, then BH n contains a Hamiltonian cycle passing through every edge of P if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths.
Vertex-disjoint paths joining adjacent vertices in faulty hypercubes
Theoretical Computer Science, 2019
Let Q n denote the n-dimensional hypercube and the set of faulty edges and faulty vertices in Q n be denoted by F e and F v , respectively. In this paper, we investigate Q n (n ≥ 3) with |F e | + |F v | ≤ n − 3 faulty elements, and demonstrate that there are two fault-free vertex-disjoint paths P [a, b] and P [c, d] satisfying that 2 ≤ (P [a, b]) + (P [c, d]) ≤ 2 n − 2|F v | − 2, where 2|((P [a, b]) + (P [c, d])), (a, b), (c, d) ∈ E(Q n). The contribution of this paper is: (1) we can quickly obtain the interesting result that Q n − F e is bipancyclic, where |F e | ≤ n − 2 and n ≥ 3; (2) this result is a complement to Chen's part result (Chen (2009) [2]) in that our result shows that there are all kinds of two disjoint-free (S, T)-paths which contain 4, 6, 8,. .. , 2 n − 2|F v | vertices respectively in Q n when S = {a, c}, T = {b, d}, and (a, b), (c, d) ∈ E(Q n). Our result is optimal with respect to the number of fault-tolerant elements.
Disjoint path covers with path length constraints in restricted hypercube-like graphs
Journal of Computer and System Sciences, 2017
A disjoint path cover of a graph is a set of pairwise vertex-disjoint paths that altogether cover every vertex of the graph. In this paper, we prove that given k sources, s 1 ,. . ., s k , in an mdimensional restricted hypercube-like graph with a set F of faults (vertices and/or edges), associated with k positive integers, l 1 ,. . ., l k , whose sum is equal to the number of fault-free vertices, there exists a disjoint path cover composed of k fault-free paths, each of whose paths starts at s i and contains l i vertices for i ∈ {1,. .. , k}, provided |F| + k ≤ m − 1. The bound, m − 1, on |F| + k is the best possible.