#### 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.

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)**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 letBy 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

**(a)**and

**(b)**and

**(a)**

**(b)**and

**(a)**

**(b)**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).