International Journal of Combinatorics
Volume 2015 (2015), Article ID 513073, 6 pages
http://dx.doi.org/10.1155/2015/513073

Research Article

Hamilton Paths and Cycles in Varietal Hypercube Networks with Mixed Faults

Jian-Guang Zhou and Jun-Ming Xu

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.

Figure 1: The varietal hypercubes , , , and .

For convenience, we write , where and . Clearly, the set of edges between and is a perfect matching of size in . Use to denote an edge in joining and . By the recursive definition of , and . Thus, is of the recursive structure shown as in Figure 2.

(a)

(b)

Figure 2: The recursive structure of .

Use and to denote two subgraphs of induced by and , respectively. It should be noted that and are not always isomorphic to , although and are isomorphic to .

Definition 2. The graph is the labeled graph defined recursively as follows. is the complete graph of two vertices labeled with and , respectively. is obtained from and plus two edges joining and , , and . For , is obtained from and by adding a perfect matching between and , according to the following rule: consists of two perfect matchings and , where is a perfect matching between and and is a perfect matching between and .

Clearly, by Definition 1, in , the set of edges between and is a perfect matching between them satisfying the rule in Definition 2. Thus, is a special example of . We state this fact as a simple observation.

Observation 1. For each , for the perfect matching defined by the rule in Definition 1. Moreover, or , where is a -dimensional cube.

3. Main Results

Let be a graph, and let and be two distinct vertices in . A subgraph of is called an -path, if its vertex-set can be expressed as a sequence of adjacent vertices, written as , in which , , and all the vertices are different from each other. For a path , we can write , and the notation denotes the subgraph obtained from by deleting the edge . If is an -path and , then is called a cycle in . A cycle is called a Hamilton cycle if it contains all vertices in . An -path is called an -Hamilton path if it contains all vertices in . A graph is Hamiltonian if it contains a Hamilton cycle and is called Hamilton-connected if it contains an -Hamilton path for any two vertices and in . Clearly, if has at least three vertices and is Hamilton-connected, then it certainly is Hamiltonian; moreover, every edge is contained in a Hamilton cycle.

Lemma 3 (Cao et al. [13]). is Hamilton-connected for , and so every edge of is contained in a Hamilton cycle for .

Let be a subset of . A subgraph of is called fault-free if contains no elements in . A graph is called -edge-fault-tolerant Hamiltonian (resp., -edge-fault-free Hamilton-connected) if contains a Hamilton cycle (resp., is Hamilton-connected) for any with . is called -fault-tolerant Hamiltonian (resp., -fault-free Hamilton-connected) if contains a Hamilton cycle (resp., is Hamilton-connected) for any with .

Lemma 4 (Huang and Xu [14]). is -edge-fault-tolerant Hamiltonian for and -edge-fault-tolerant Hamilton-connected for .

In this paper, we will generalize this result by proving that is -fault-tolerant Hamiltonian for and -fault-tolerant Hamilton-connected for .

To prove our main results, we first prove the following result on the graph .

Theorem 5. For , is -fault-tolerant Hamilton-connected for any perfect matching between and defined by the rule in Definition 2.

Proof. We proceed by induction on .
Since or , which is vertex-transitive, it is easy to check the conclusion is true for . Suppose now that and the result holds for any integer less than . Let with , and let and be two distinct vertices in . We need to prove that contains an -Hamilton path. Without loss of generality, we can assume . Let , where

and let

By symmetry of structure of , we may assume .
Case 1 ( ). In this case, by the hypothesis, we have .
Subcase 1.1 ( or ). Without loss of generality, assume .
Since and , by the induction hypothesis contains an -Hamilton path, say . Since , there is an edge in such that the neighbors and of and in are not in . Since and , by the induction hypothesis contains a -Hamilton path, say . Thus, is an -Hamilton path in (see Figure 3(a)).
Subcase 1.2 ( and ). Since and , there is an edge such that and are not in . By the induction hypothesis, let be an -Hamilton path in , and let be a -Hamilton path in . Then is an -Hamilton path in (see Figure 3(b)).
Case 2 ( ). In this case, .
Subcase 2.1 ( ). Arbitrarily take a vertex . Since , by the induction hypothesis contains an -Hamilton path, say . Without loss of generality, assume . Let and be two neighbors of in , and let . By the induction hypothesis, contains a -Hamilton path, say . Then is an -Hamilton path in .
Subcase 2.2 ( and ). If , then or . Since and is vertex-transitive, we can assume unless . It is easy to check that contains a Hamilton cycle, say . Choose a neighbor of in such that its neighbor in is not . By the induction basis, contains a -Hamilton path, say . Then, is an -Hamilton path in .
Assume now ; that is, . Let . Without loss of generality, we can assume .
(a) (See Figure 4(a)). Arbitrarily take with , and let . Since , by the induction hypothesis contains a -Hamilton path, say . Arbitrarily take a vertex . Since , by the induction hypothesis contains an -Hamilton path, say . If is in , then let and be two neighbors of in ; if is not in , then let be an edge in . Let . By the induction hypothesis, contains a -Hamilton path, say . Let if is in and if is not in . Then is an -Hamilton path in (see Figure 4(a)).
(b) (See Figure 4(b)). Arbitrarily take a vertex in with . Let be the neighbor of in . Arbitrarily take a vertex . Since , by the induction hypothesis contains an -Hamilton path, say . If is in , then let and be two neighbors of in ; if is not in , then let be an edge in . Let . By the induction hypothesis, contains a -Hamilton path, say . Since , we can write . Let be the neighbor of in . By the induction hypothesis, contains a -Hamilton path, say . Let if is in and if is not in . Then is an -Hamilton path in (see Figure 4(b)).
Subcase 2.3 ( ). If , then . By the induction basis, contains an -Hamilton path, say . Since is vertex-transitive and , it is easy to check that contains a Hamilton cycle, say . Since and are -regular and isomorphic, there is an edge in which is not incident with and such that the corresponding edge in is contained in . By Definition 2, where and are neighbors of and in , respectively. Thus, is an -Hamilton path in (as a reference, see Figure 3(a)).
Assume below; that is, .
(a) (See Figure 5(a)). By the induction hypothesis, contains an -Hamilton path, say . Take , and let and be neighbors of and in , respectively. Take a vertex in . By the induction hypothesis, contains a -Hamilton path, say . If is in , then let and be two neighbors of in ; if is not in , then let be an edge in . Let and be neighbors of and in , respectively. By the induction hypothesis, contains a -Hamilton path, say . Let if is in and if is not in . Thus, is an -Hamilton path in (see Figure 5(a)).
(b) and (See Figure 5(b)). Arbitrarily take a vertex in and an edge in . By the induction hypothesis, contains a -Hamilton path, say . If is in , then let ; if is not in , then let . Without loss of generality, assume that is in and let and be two neighbors of in .
Let and be neighbors of and in , respectively. By the induction hypothesis, contains a -Hamilton path, say . Since is in , we can write (see Figure 5(b)). Let be the neighbor of in . By the induction hypothesis, contains an -Hamilton path, say . Then is an -Hamilton path in (see Figure 5(b)).
(c) (See Figure 6)
(c1) . By the induction hypothesis, contains an -Hamilton path, say . Take , and let and be neighbors of and in , respectively. Take a vertex in . By the induction hypothesis, contains a -Hamilton path, say . If is in , let and be two neighbors of in ; if is not in , let be an edge in . Let if is in and if is not in .
Let and be neighbors of and in , respectively. By the induction hypothesis, contains a -Hamilton path, say . Thus, is an -Hamilton path in (see Figure 6(a)).
(c2) . In this case, since . Consider the subgraph of induced by . By Definition 2, it is easy to check that . Let . By the induction hypothesis, contains an -Hamilton path, say . Without loss of generality, assume that is in . Let and be two neighbors of in , and let and be two neighbors of and in . Then there is a -Hamilton path in , say . Take an edge in , and let and be neighbors of and in . Then there is a -Hamilton path in , say . Thus, is an -Hamilton path in (see Figure 6(b)).
The theorem follows.

(a)

(b) and

Figure 3: Illustrations of Case 1 in the proof of Theorem 5.

(a) and

(b) and

Figure 4: Illustrations of Subcase 2.2 in the proof of Theorem 5.

(a)

(b) and

Figure 5: Illustrations of Subcase 2.3 in the proof of Theorem 5.

(a)

(b)

Figure 6: Illustrations of Subcase 2.3(c) in the proof of Theorem 5.

By Observation 1 and Theorem 5, we have the following results immediately.

Corollary 6. is -fault-tolerant Hamilton-connected for .

Corollary 7. Every fault-free edge of is contained in a fault-free Hamilton cycle if the number of faults does not exceed and .

Proof. If , then the conclusion holds clearly. Assume now . Let be a fault-free edge in . Let be a set of faults in with and containing the edge . By Corollary 6, there is an -Hamilton path in . Then is a required cycle.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The work was supported by NNSF of China (no. 61272008).

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