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

Abstract

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.

The bicube: an interconnection of two hypercubes

“…The hamiltonicity and pancyclicity of HL-graphs were considered in [19] as follows. For more discussion on hamiltonicity and its generalization, refer to [15,18]. It is obvious that BQ m is a bipartite HL-graph, so the lemma follows: Lemma 4.10 BQ m is hamiltonian-laceable and bipancyclic for m ≥ 3.…”

Many-to-many two-disjoint path covers in restricted hypercube-like graphs

“…Given S and T in a graph G, it is NP-complete to determine if there exists a one-to-one, one-to-many, or many-to-many k-DPC joining S and T for any fixed k ≥ 1 [32,33]. The disjoint path cover problems have been studied for graphs such as hypercubes [5,6,7,10,13,19,24], recursive circulants [20,21,32,33], and hypercube-like graphs [18,22,28,33], cube of a connected graph [29,30], and k-ary n-cubes [35,37]. Necessary conditions for a graph G to be f -fault many-to-many k-disjoint path coverable have been established in terms of its connectivity κ(G) and its minimum degree δ(G) [32,33], as shown below.…”