Survey on path and cycle embedding in some networks

Abstract

To find a cycle (resp. path) of a given length in a graph is the cycle (resp. path) embedding problem. To find cycles of all lengths from its girth to its order in a graph is the pancyclic problem. A stronger concept than the pancylicity is the panconnectivity. A graph of order n is said to be panconnected if for any pair of different vertices x and y with distance d there exist xy-paths of every length from d to n. The pancyclicity or the panconnectivity is an important property to determine if the topology of a network is suitable for some applications where mapping cycles or paths of any length into the topology of the network is required. The pancyclicity and the panconnectivity of interconnection networks have attracted much research interest in recent years. A large amount of related work appeared in the literature, with some repetitions. The purpose of this paper is to give a survey of the results related to these topics for the hypercube and some hypercube-like networks.

Long cycles in hypercubes with optimal number of faulty vertices

“…For the completeness, let us also mention that there are many related results on similar problems of bipanconnectivity, bipancyclicity, long cycles, and long paths in various modifications of faulty hypercubes, see a survey of Xu and Ma [14] for further references.…”

Unpaired many-to-many vertex-disjoint path covers of a class of bipartite graphs

“…There is a large amount of literature on graph-theoretical properties of the n-cube and their applications in parallel computing, e.g., see [9,14], a recent survey [15] and references therein. The n-cube is a graph with 2 n vertices, its any vertex is denoted by an n-bit binary string.…”

Hamilton Paths and Cycles in Varietal Hypercube Networks with Mixed Faults

“…Embedding paths and cycles in various well-known networks, such as the hypercube and some well-known variations of the hypercube, have been extensively investigated in the literature (see, e.g., Tsai [5] for the hypercubes, Fu [6] for the folded hypercubes, Huang et al [7] and Yang et al [8] for the crossed cubes, Yang et al [9] for the twisted cubes, Hsieh and Chang [10] for the Möbius cubes, Li et al [11] for the star graphs and Xu and Ma [12] for a survey on this topic). Recently, Cao et al [13] have shown that every edge of is contained in cycles of every length from 4 to 2 except 5, and every pair of vertices with distance is connected by paths of every length from to 2 − 1 except 2 and 4 if = 1, from which contains a Hamilton cycle for ⩾ 2 and a Hamilton path between any pair of vertices for ⩾ 3.…”