Hamilton Paths and Cycles in Varietal Hypercube Networks with Mixed Faults

and proposed by Cheng and Chuang [1] in 1994 and has many properties similar or superior to Q n . 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, faultdiameter, and wide-diameter of VQ n are smaller than those of the hypercube Q n (see Cheng and Chuang [1], Jiang et al. [3]). Recently, Xiao et al. [4] have shown that VQ n is vertextransitive. 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 VQ n is contained in cycles of every length from 4 to 2n except 5, and every pair of vertices with distance d is connected by paths of every length from d to 2n − 1 except 2 and 4 if d = 1, from which VQ n contains a Hamilton cycle for n ⩾ 2 and a Hamilton path between any pair of vertices for n ⩾ 3. Huang and Xu [14] have improved this result by considering edge-faults and showing that VQ n contains a fault-free Hamilton cycle provided faulty edges do not exceed n − 2 for n ⩾ 3 and a fault-free Hamilton path between any pair of vertices provided faulty edges do not exceed n − 3 for n ⩾ 3. In this paper, we will further improve these results by considering mixed faults of vertices and edges and proving that VQ n contains a fault-free Hamilton cycle provided the number of mixed faults does not exceed n − 2 for n ⩾ 2 and contains a fault-free Hamilton path between any pair of vertices provided the number of mixed faults does not exceed n − 3 for n ⩾ 3. The proofs of these results are in Section 3.The definition and some basic structural properties of VQ n are given in Section 2.


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, faultdiameter, 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 vertextransitive.
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. 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  − 2 for  ⩾ 3 and a fault-free Hamilton path between any pair of vertices provided faulty edges do not exceed  − 3 for  ⩾ 3.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  − 2 for  ⩾ 2 and contains a fault-free Hamilton path between any pair of vertices provided the number of mixed faults does not exceed  − 3 for  ⩾ 3.
The proofs of these results are in Section 3. The definition and some basic structural properties of   are given in Section 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 vertexset and  = () is the edge-set of .For  ∈ (), we call  (resp., ) a neighbor of  (resp., ).
−1 and  1 −1 by adding a perfect matching  between  0 −1 and  1 −1 , according to the following rule:  consists of two perfect matchings  1 and  2 , where  1 is a perfect matching between  00 −2 and  10 −2 and  2 is a perfect matching between  01 −2 and  11 −2 .
Clearly, by Definition 1, in   , the set  of edges between  0 −1 and  1 −1 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.
for the perfect matching  defined by the rule in Definition 1.Moreover,  3 ≅  3 or  3 , where  3 is a 3-dimensional cube.

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  = ( 0 ,  1 ,  2 , . . .,   ), in which  =  0 ,  =   , and all the vertices  0 ,  Lemma 3 (Cao et al. [13]).  is Hamilton-connected for  ⩾ 3, and so every edge of   is contained in a Hamilton cycle for  ⩾ 2.
To prove our main results, we first prove the following result on the graph   .Proof.We proceed by induction on  ⩾ 3.
Since  3 ≅  3 or  3 , which is vertex-transitive, it is easy to check the conclusion is true for  = 3. Suppose now that  ⩾ 4 and the result holds for any integer less than .Let  ⊂ (  ) ∪ (  ) with || ⩽  − 3, 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 (|  | ⩽  − 4).In this case, by the hypothesis, we have        path, say   .If  is in   , then let  00 and  00 be two neighbors of  in   ; if  is not in   , then let  00 V 00 be an edge in   .

Subcase 2.3 (𝑥, 𝑦 ∈ 𝑅).
If  = 4, then  ≅  ≅  3 .By the induction basis,  contains an -Hamilton path, say   .Since  3 is vertex-transitive and |  | = 1, it is easy to check that −  contains a Hamilton cycle, say   .Since  and  are 3-regular and isomorphic, there is an edge   V  in   which is not incident with  and  such that the corresponding edge   in  is contained in   .By Definition 2   =   V  , where   and V  are neighbors of   and V  in , respectively.Thus,   −   V  +     + V  V  +   −   is an -Hamilton path in  4 −  (as a reference, see Figure 3(a)).
(b)  ∈  11  −1 and  ∈  10 −2 (See Figure 5(b)).Arbitrarily take a vertex  in  00 and an edge  00 V 00 in  00 −2 .By the induction hypothesis,  − (  − ) contains a  00 V 00 -Hamilton path, say   .If  is in   , then let   =   −  +  00 V 00 ; if  is not in   , then let   =   .Without loss of generality, assume that  is in   and let  00 and V 00 be two neighbors of  in   .
Let  10 and V 10 be neighbors of  00 and V 00 in  10 −2 , respectively.By the induction hypothesis,  10  −2 contains a  10 V 10 -Hamilton path, say  10 .Since  is in  10 , we can write  10 =  10 (V 10 , ) +  10 +  10 ( 10 ,  10 ) (see Figure 5      The theorem follows. By Observation 1 and Theorem 5, we have the following results immediately.Proof.If  = 2, then the conclusion holds clearly.Assume now  ⩾ 3. Let  be a fault-free edge in   .Let  be a set of faults in   with || ⩽ −2 and containing the edge .By Corollary 6, there is an -Hamilton path  in   −(−).Then  +  is a required cycle.

Figure 2 :
Figure 2: The recursive structure of   .
Hamilton-connected for any perfect matching  between  0 −1 and  1 −1 defined by the rule in Definition 2.

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

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

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

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

Corollary 6 .Corollary 7 .
is (−3)-fault-tolerant Hamilton-connected for  ⩾ 3. Every fault-free edge of   is contained in a fault-free Hamilton cycle if the number of faults does not exceed  − 2 and  ⩾ 2.