International Journal of Combinatorics

Volume 2015 (2015), Article ID 513073, 6 pages

http://dx.doi.org/10.1155/2015/513073

## Hamilton Paths and Cycles in Varietal Hypercube Networks with Mixed Faults

Department of Mathematics, University of Science and Technology of China, Wentsun Wu Key Laboratory of CAS, Hefei, Anhui 230026, China

Received 16 September 2014; Accepted 5 January 2015

Academic Editor: Chris A. Rodger

Copyright © 2015 Jian-Guang Zhou and Jun-Ming Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper considers the varietal hypercube network with mixed faults and shows that contains a fault-free Hamilton cycle provided faults do not exceed for and contains a fault-free Hamilton path between any pair of vertices provided faults do not exceed for . The proof is based on an inductive construction.

#### 1. Introduction

As a topology of interconnection networks, the hypercube is the most simple and popular since it has many nice properties. The varietal hypercube is a variant of and proposed by Cheng and Chuang [1] in 1994 and has many properties similar or superior to . For example, they have the same numbers of vertices and edges and the same connectivity and restricted connectivity (see Wang and Xu [2]), while all the diameter and the average distances, fault-diameter, and wide-diameter of are smaller than those of the hypercube (see Cheng and Chuang [1], Jiang et al. [3]). Recently, Xiao et al. [4] have shown that is vertex-transitive.

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 except 5, and every pair of vertices with distance is connected by paths of every length from to except and if , from which contains a Hamilton cycle for and a Hamilton path between any pair of vertices for . Huang and Xu [14] have improved this result by considering edge-faults and showing that contains a fault-free Hamilton cycle provided faulty edges do not exceed for and a fault-free Hamilton path between any pair of vertices provided faulty edges do not exceed for . In this paper, we will further improve these results by considering mixed faults of vertices and edges and proving that contains a fault-free Hamilton cycle provided the number of mixed faults does not exceed for and contains a fault-free Hamilton path between any pair of vertices provided the number of mixed faults does not exceed for .

The proofs of these results are in Section 3. The definition and some basic structural properties of are given in Section 2.

#### 2. Definitions and Structural Properties

We follow [15] for graph-theoretical terminology and notation not defined here. A graph always means a simple and connected graph, where is the vertex-set and is the edge-set of . For , we call (resp., ) a neighbor of (resp., ).

Let be a labeled graph with vertex set . For , let , where for each . Use to denote a labeled graph obtained from by inserting the string in front of each vertex-labeling in . Clearly, .

*Definition 1. *The -dimensional varietal hypercube is the labeled graph defined recursively as follows. is the complete graph of two vertices labeled with and , respectively. Assume that has been constructed. For , is obtained from and by joining vertices between them, according to the rule: a vertex in and a vertex in are adjacent in if and only if(1) if , or(2) and if , where .

Figure 1 shows the examples of varietal hypercubes for , and , respectively.