1. Introduction
In massive processing systems (MPS), processors are connected based on a specific interconnection network topology, which is usually represented by an undirected graph: vertices represent processors and edges represent links between processors. It is wellknown that the network topology plays a crucial role in its performance. The wellknown hypercube (or cube), which has received much attention over past decades, is one of the most popular topologies for MPS. The hypercube possesses numerous desirable properties for MPS, such as large bandwidth, short (logarithmic) diameter, high symmetry and connectivity, recursive structure, simple routing and broadcasting. With so many advantages, a number of hypercube machines have been implemented [14], such as Cosmic Cube [30], the Ametek S/14 [6], the iPSC [12], the NCUBE [27], and the CM200 [9].
However, a network topology can not be optimum in all aspects. To enhance some properties of the hypercube, a number of variations of the hypercube have been proposed. One method of reducing diameter of the hypercube is to still retain the “hypercubelike” structure by twisting some pair of edges, such as twisted cube [1], Möbius cube [10], crossed cube [17] and Mcube [31]. Another feasible approach is to reduce vertex degree that leads to technological problems in parallel computing, such as Fibonacci cube [16], Lucas cube [26] and exchanged hypercube [25]. In addition, other pleasing hypercube variants are obtained by adopting different techniques, such as balanced hypercube [32], folded hypercube [13], generalized hypercube [7] and shufflecube [20]. The shufflecube has some good combinatorial properties and faulttolerant properties. The reliability of the shuffle cube concerning refined connectivity were determined in [33, 29, 11]. The conditional diagnosability of was studied in [34, 23]. The matching preclusion number was determined by Antantapantula et al. [4]. Faulttolerant Hamiltonian cycle embedding of the shufflecube was investigated by Li et al. [21].
To achieve high performance, graphs with high levels of symmetry (e.g., Cayley graphs [2, 19]) are recommended as network topologies since it often simplifies the computation and routing algorithms. Actually, vertextransitive and/or edgetransitive graphs are widely used to design networks of high levels of symmetry. For example, numerous attractive networks, including hypercube, ary cube, balanced hypercube, folded hypercube, star graphs and some of their variants, which have both theoretical and practical importance, are vertextransitive and/or edgetransitive. On the other hand, to study the symmetry of a graph, the aim is to obtain as much information as possible about its symmetric property.
Routing is the problem of finding path between each pair of vertices in a graph. It is wellknown that a good routing algorithm does not require large amount of memory resources to build the routing table, and its convergence time is usually slow for large networks. Because of the importance of a good routing in interconnection networks, it has received considerable attention in the literature [22, 24, 18, 35]. In particular, a path/cycle consisting of all nodes of a network, i.e. Hamiltonian path/cycle, has a wide range of importance in theory and practice since there are some applications of Hamiltonian path in the online optimization of a complex Flexible Manufacturing System [5], as well as full utilization of all nodes in a network [28].
The rest of this paper is organized as follows. In Section 2, some notations and the definition of the shufflecube are presented. In Section 3, it is showed that the shufflecube is neither vertextransitive nor edgetransitive. In Section 4, two vertextransitive variants of the shuffle cube are proposed. The routing algorithms and Hamiltonian cycle embeddings of the variants are investigated in Section 5 and 6, respectively. Conclusions are given in Section 7.
2. Preliminaries
Let be a graph, where is the vertexset of and is the edgeset of . The number of vertices of is denoted by . A path in is a sequence of distinct vertices so that there is an edge joining each pair of consecutive vertices. If a path is such that , , then is said to be a cycle, and the length of is the number of edges contained in . The length of the shortest cycle of is called the girth of , denoted by . In particular, a cycle containing all vertices of is called a Hamiltonian cycle. The clique of is a set of pairwise adjacent vertices and the clique number of , is the maximum size of a clique in . A graph is vertextransitive if for each pair there exists an automorphism that maps to . A graph is edgetransitive if for all there exists an automorphism of that maps the endpoints of to the endpoints of . For other standard graph notations not defined here please refer to [8].
The vertices are labelled by binary sequences of bit. For a vertex , for each , the prefix of is , written by , and the suffix of is , written by . The Hamming distance of two vertices and , denoted by , is the number of bits which they differ. The wellknown dimensional hypercube , consists of all of the bit binary sequences as its vertex set and two distinct vertices and are linked by an edge if and only if .
To recursively build shufflecubes, we define four sets containing tuple of binary sequences as follows:

,

,

,

.
We are ready to give the definition of the shufflecube.
Definition 1
.[20] The dimensional shufflecube, , is recursively defined as follows: . For , contains exactly 16 subcubes , where and all vertices of share the same . The vertices and in different subcubes of dimension are adjacent in iff

, and

,
where the notation “” means bitwise addition under modulo 2.
By the definition above, it is clear that is regular and , which is the same as that of . For clarity, is illustrated in Fig. 1 with only edges incident to vertices in .
By Definition 1, it implies that two vertices in different subcubes of dimension differ in exactly one 4bit. For convenience, the th 4bit of a vertex , denoted by , is defined as , . For notation consistency, we define . For two distinct vertices and in , if they bitwise equal. The complementary of an arbitrary bit of , i.e. , is denoted by . Similar to Hamming distance, we define 4bit Hamming distance between and , written by , as the number of 4bits with such that . In particular, we use to denote the number of 4bits with such that .
3. Symmetric properties of shufflecubes
In this section, we shall state some symmetric properties of shufflecubes. We begin with the following lemma.
Lemma 1
. is nonbipartite for all . Moreover, for all .
Proof. To show that is nonbipartite for all
, it suffices to present an odd cycle of
. Let be a vertex in with and let and be four neighbors of with for some . It follows that .Additionally, we may assume that and . It follows from Definition 1 that and are pairwise adjacent, forming a clique of size four. Moreover, is not adjacent to none of and . Suppose that there exists another vertex () such that is adjacent to all of and . We claim that . Otherwise, suppose without loss of generality that . Therefore, for some , . So and (resp. , ) differ in two 4bits, a contradiction. Thus, the claim holds. Note that , then . By Definition 1, is not adjacent to and , a contradiction again.
This completes the proof. ∎
Lemma 2
. A vertex of of () is contained in a clique of size four if and only if . Moreover, the clique number of is four.
Proof. It has been shown in the proof of Lemma 1 that if , there are four vertices and forming a clique of size four. To prove the clique number of is four, we show that there exists no vertex adjacent to all of and .
In what follows, we prove the necessity. Suppose on the contrary that . We shall prove that any pair of the neighbors of are nonadjacent. Let and be two arbitrary neighbors of and let and , where . The following two cases arise.
Case 1: . This implies that and differ in two bits. By Definition 1, and are nonadjacent.
Case 2: . If , then and are in the same of . By Definition 1, and are nonadjacent. So we assume that . Clearly, . It follows that and , implying that . On the contrary, suppose that and are adjacent. Since , we have , where are two bits we do not care. This contradicts the fact that the first two bits of , are the same as . This completes the proof. ∎
Remark 1. Any vertex of () with and its bit neighbors and with , form a clique of size four. So is contained in exactly cliques of size four. Moreover, consists of vertexdisjoint clique of size four.
Theorem 3
. is vertex transitive for , but not vertextransitive for all .
Proof. Clearly, consists of 16 vertexdisjoint 4cycles, which is obviously vertextransitive. By Lemma 2, we know that each vertex with is contained in a clique of size four in , . Any vertex with is contained in no clique of size four, showing obviously that is not vertextransitive for all . ∎
It is wellknown that an edgetransitive but not vertextransitive graph must be bipartite. Combing Lemma 1 with Theorem 3, we have the following corollary.
Corollary 4
. is not edgetransitive for all .
4. Two vertextransitive variations of the shufflecubes
As we have shown in the previous section, the shufflecube () is not vertextransitive. In this section, we shall present two variants of the shufflecubes, which are both vertextransitive. We shall adopt the notations which we have defined in the .
Definition 2
. The dimensional simplified shuffle cube, , is recursively defined as follows: . For , contains exactly 8 subcubes , where , and all vertices of share the same . The vertices and in different subcubes of dimension are adjacent in iff

, and

,
where the notation “” means bitwise addition under modulo 2.
The reason why we define (which is slightly different from the definition of ) is that if we define as in Definition 1, then the resulting graph is always disconnected, which does not meet the basic requirement of interconnection networks. As a result, has vertices, . Clearly, is also regular and nonbipartite. is illustrated in Fig. 2 with only edges incident to vertices in .
In what follows, we shall prove the vertextransitivity of , .
Theorem 5
. is vertextransitive whenever .
Proof. Obviously, is vertextransitive if . It suffices to show that is vertextransitive for .
Let and be any two distinct vertices of . Our objective is to show that there is an automorphism of such that . Thus, for any vertex of , we define 4bit by 4bit (in fact, contains exactly two bits) below:
Now we are ready to show that is an isomorphism of . Let be an arbitrary edge of . We shall show that is also an edge of . By Definition 2, we known that and differ in exactly one 4bit (including the last two bits), say , .
If , then and . By the definition of , we have and . Additionally, for . Thus, .
If for some , then and for all .
By the definition of , we have and . Additionally, for all . Thus, . This completes the proof.∎
In the following, we shall give another variation of the shuffle cubes, which is motivated by the idea of construction of the balanced hypercubes [32]. For the readability of this paper, we first present the definition of the balanced hypercube, and then give the definition of the balanced shuffle cube.
Definition 3
[32]. An dimensional balanced hypercube contains vertices , where . Any vertex in has the following neighbors:

mod ,
mod , and 
mod mod ,
mod mod .
Definition 4
. The dimensional balanced shuffle cube, , is recursively defined as follows: . For , contains exactly 16 subcubes , where and all vertices of share the same . The vertices and in different subcubes of dimension are adjacent in iff

, and

and have different parities, and , and

, or .
where addition and substraction are under modulo 4 by regarding the two bits as an integer.
Clearly, has vertices, . Additionally, is regular and bipartite. is illustrated in Fig. 3 with only edges incident to vertices in .
We shall prove the following statement, which directly leads to the regularity of , .
Theorem 6
. is vertextransitive whenever .
Proof. Clearly, is vertextransitive when . It remains to show that is vertextransitive for .
Let and be any two distinct vertices of . Our aim is to show that there is an automorphism of such that . Thus, for any vertex of , we define 4bit by 4bit (in fact, contains exactly two bits) below:
If both of and are even, then
If is even and is odd, then
If is odd and is even, then
If both of and are odd, then
, and addition and substraction are under modulo 4 by regarding the two bits as an integer. In addition,
Let be an arbitrary edge of with and . We shall show that is also an edge of . By Definition 4, it is obvious that and differ in exactly one 4bit (including the last two bits), say , .
If , then and . By the definition of , we have and . Additionally, for . Thus, .
If for some , then and for all . By the definition of , the following cases arise.
Case 1. is even.
Case 1.1. is even. Clearly, we have
and
. Thus, the parity of (resp. ) is the same as that of (resp. ). It follows from Definition 4 that .
Case 1.2. is odd. Clearly, we have
and
. Observe that .
If is even, then . Thus, . So . Now we see that is odd and is even. By Definition 4, we have .
If is odd, then . Thus, . So . Now we see that is even and is odd. By Definition 4, we have .
Case 2. is odd.
Case 2.1. is even. In this case, we have
and
. Obviously, and have different parities. Observe that .
If is even, then . Thus, . Now we see that is odd and is even. By Definition 4, we have .
If is odd, then . Thus, . Now we see that is even and is odd. By Definition 4, we have .
Case 2.2. is odd. In this case, we have
, and
.
If is even, then