Fault-Tolerant Hamiltonian Laceability and Fault-Tolerant Conditional Hamiltonian for Bipartite Hypercube-Like Networks

Abstract

A bipartite graph G is hamiltonian laceable if there is a hamiltonian path between any two vertices of G from distinct vertex bipartite sets. A bipartite graph G is k-edge fault-tolerant hamiltonian laceable if G - F is hamiltonian laceable for every F ⊆ E(G) with |F| ≤ k. A graph G is k-edge fault-tolerant conditional hamiltonian if G - F is hamiltonian for every F ⊆ E(G) with |F| ≤ k and δ(G - F) ≥ 2. Let G0 = (V0, E0) and G1 = (V1, E1) be two disjoint graphs with |V0| = |V1|. Let Er = {(v,ɸ(v)) | v ϵ V0,ɸ(v) ϵ V1, and ɸ: V0 → V1 is a bijection}. Let G = G0 ⊕ G1 = (V0 ⋃ V1, E0 ⋃ E1 ⋃ Er). The set of n-dimensional hypercube-like graphHn is defined recursively as (a) H1 = K2, K2 is the complete graph with two vertices, and (b) if G0 and G1 are in Hn, then G = G0 ⊕ G1 is in Hn+1. Let Bn be the set of graphs G where G is bipartite and G ϵ Hn. In this paper, we show that every graph in Bn is (n - 2)-edge fault-tolerant hamiltonian laceable if n ≥ 2 and every graph in Bn is (2n - 5)-edge fault-tolerant conditional hamiltonian if n ≥ 3.

Paired 2-disjoint path covers and strongly Hamiltonian laceability of bipartite hypercube-like graphs

“…Theorem 3 is an extension of the work in [21,26], which is stated in Lemma 2. The total number m − 2 of faulty elements in Theorem 3 is the maximum possible regardless of whether there exists a faulty vertex or not.…”

“…Hamiltonian-laceability of X m with at most m − 2 faulty edges was studied independently by Park [26] and by Lin et al [21], as shown in Lemma 2. In addition, a Hamiltonian property of X m in the presence of a single vertex fault was reported in [22,26], as shown in Lemma 3.…”

“…For the problem of Hamiltonian paths passing through prescribed edges, see [6,32] for example. Furthermore, disjoint path coverability has been employed to establish some Hamiltonian properties, such as in hypercube-like graphs [21,27,29].…”

“…Every m-dimensional bipartite HL-graph X m , m ≥ 2, has a paired 2-DPC joining S and T if S and T are subsets of different parts [27]. Every X m , m ≥ 2, possessing at most m − 2 faulty edges is Hamiltonian-laceable [21,26]. An equitable bipartite graph is called Hamiltonian-laceable if each pair of vertices contained in different parts is joined by a Hamiltonian path.…”