1. Introduction
The interconnection network (network for short) plays an important role in massively parallel and distributed systems [12]. Linear arrays and rings are two fundamental networks. Since some parallel applications such as those in image and signal processing are originally designated on an array architecture, it is important to have effective path embedding in a network [1, 20]. To find parallel paths among vertices in networks is one of the most central issues concerned with efficient data transmission [12]. Parallel paths in networks are usually studied with regard to disjoint paths in graphs. Moreover, algorithms designed on linear arrays or rings can be efficiently simulated in a topology containing paths or cycles, so path and cycle embedding properties of networks have been widely studied [3, 5, 6, 7, 8, 10, 11, 18, 19, 22, 23].
In disjoint path cover problems, the manytomany disjoint path cover problem is the most generalized one[17]. Assume that and are two sets of sources and sinks in a graph , respectively, the manytomany disjoint path cover (DPC for short) problem is to determine whether there exist disjoint paths in such that joins to for each and , where is a permutation on the set . The DPC is called paired if is the identical permutation and unpaired otherwise. Interestingly, the DPC problem is closely related to the wellknown Hamiltonian path problem in graphs. In fact, a 1DPC of a network is indeed a Hamiltonian path between any two vertices.
The performance of the famous hypercube network is not optimum in all aspects, accordingly, many variants of the hypercube have been proposed. The balanced hypercube, proposed by Wu and Huang [21], is such the one of the most popularity. The special property of the balanced hypercube, which other hypercube variants do not have, is that each processor has a backup processor that shares the same neighborhood. Thus tasks running on a faulty processor can be shifted to its backup one [21]. With such novel properties above, different aspects of the balanced hypercube were studied extensively, including path and cycle embedding issues [9, 13, 16, 22, 23, 25], connectivity [15, 24], matching preclusion[14], and symmetric properties [26, 27].
Recently, Cheng el al. [4] proved that the balanced hypercube () has a paired 2DPC, which is a generalization of Hamiltonian laceability of the balanced hypercube [22]. To the best of our knowledge, there is no literature on DPC in the balanced hypercube when . In this paper, we will consider the problem of unpaired DPC of the balanced hypercube.
The rest of this paper is organized as follows. In Section 2, some definitions and lemmas are presented. The main result of this paper is shown in Section 3. Conclusions are given in Section 4.
2. Preliminaries and some lemmas
A network is usually modeled by a simple undirected graph, where vertices represent processors and edges represent links between processors. Let be a graph, where and are its vertexset and edgeset, respectively. The number of vertices of is denoted by . The set of vertices adjacent to a vertex is called the neighborhood of , denoted by . A path in is a sequence of distinct vertices so that there is an edge joining each pair of consecutive vertices. If and , then the graph is called a cycle. A path (resp. cycle) containing all vertices of a graph is called a Hamiltonian path (resp. cycle). A graph admits a Hamiltonian cycle is a Hamiltonian graph. A Hamiltonian bipartite graph is Hamiltonian laceable if, for any two vertices and from different partite sets, there exists a Hamiltonian path between and . For other standard graph notations not defined here please refer to [2].
The definitions of the balanced hypercube are given as follows.
Definition 1
.[21] An dimension balanced hypercube contains vertices , where . Any vertex in has the following neighbors:

mod ,
mod , and 
mod mod ,
mod mod .
The first coordinate of the vertex in is defined as the inner index, and are dimensional index.
The recursive definition of the balanced hypercube is presented as follows.
Definition 2
is shown in Fig. 1 (a). The standard layout of is shown in Fig. 1 (b) and the ringlike layout is shown in Fig. 1 (c).
The following basic properties of the balanced hypercube will be applied in the main result of this paper.
Lemma 1
[21]. is bipartite.
By Lemma 1, we present a bipartition and of , where contains all vertices of with even inner index, and contains all vertices of with odd inner index.
Lemma 3
[21]. Vertices and mod 4, in have the same neighborhood.
Assume that is a neighbor of in . If and differ only from the inner index, then is called a dimension edge, and and are mutually called 0dimension neighbors. Similarly, if and differ from the th outer index (), is called a dimension edge, and and are mutually called dimension neighbors. The set of all dimension edges of is denoted by for each , and the subgraph of obtained by deleting is written by , where . Obviously, each of is isomorphic to .
Lemma 4
[13]. The balanced hypercube is Hamiltonian laceable for all .
Lemma 5
[4]. Let and . Then there exist two vertexdisjoint paths and such that: (1) connects to , (2) connects to , (3) .
3. Main results
Because of the recursive structure of the balanced hypercube, we use induction to prove the main result. We start with the following useful notation.
Let and such that and , . For convenience, let denote the dimensional index of the vertex , where and .
Lemma 6
. There exists a dimension such that by splitting () along dimension , for each .
Proof.
We first consider . On the contrary, suppose that for each , by splitting along dimension , there exists some () such that . Clearly, . Then () takes at most two values. If one of , say , takes exact one value, combining with , then two vertices in have the same coordinates, which is a contradiction. So we assume that there are two values of for each . Then three of take one common value for . Thus, two of and , say and , have the same 1dimensional and 2dimensional indices, and distinct inner indices. Observe that exact one of the 1dimensional and 2dimensional indices of (resp. ) is different from that of and . Thus, by splitting along dimension , and are in the same for some , and .
Now we consider for . Clearly, . Suppose on the contrary that () take at most two values and of which take one common value. We can consider the coordinates (except inner index) of vertices in
as row vectors, forming a
matrix . Thus, there exists at least three equal rows of , indicating that there are three vertices in differing only the inner indices. Note that the inner indices of vertices in take only two values. So there are two vertices in with the same coordinates, which is a contradiction. This completes the proof. ∎By the above lemma, there exists a dimension , such that by splitting along dimension , each contains at most vertices in , . We may assume that in the rest paper. Let and and let . We have the following lemma.
Lemma 7
. There exists some such that and . Furthermore, .
Proof.
By the ringlike structure of , it is obvious that there exists some such that and , implying that and . Obviously, if for each , we are done. Next we distinguish the following cases.
Case 1. There exists exactly one integer such that or . We only consider since the same argument applies to . Clearly, and . Noting , and , it follows that .
Case 2. There exist exactly two distinct integers such that and , where . By the ringlike structure of , there are two essentially distinct cases by relative positions of and . We further distinguish the following two cases.
Case 2.1. . That is, and , and and . Obviously, .
Case 2.2. . That is, and , and and . Similarly, or holds. This completes the proof. ∎
Theorem 8
. There exists an unpaired DPC joining and .
Proof.
We prove by induction on . By Lemma 5, the theorem obviously holds for . Suppose the statement holds for with . Next we consider . By Lemmas 6 and 7, we split into four s along dimension such that for each , and for some , say , we have , and . We distinguish the following cases.
Case 1. or for each .
Case 1.1 . Choose white vertices from to generate a set . By the induction hypothesis, we can obtain DPC of from to . Note that if . Let and let such that is an dimensional neighbor of for each . Since , we have . We consider the following two conditions in terms of the cardinality of (This argument will be used repeatedly in the remaining proof).

If , then we choose white vertices from to generate a set . By the induction hypothesis, we can obtain DPC of from to . Note also that if . Let and let such that is an ()dimensional neighbor of for each .

If , then we choose white vertices from to generate a set . We then arbitrarily choose vertices from to form a set . By the induction hypothesis, we can obtain DPC of from to . Note that if . Since , there are at most two vertices of on at most two paths of DPC of . Suppose without loss of generality that . Additionally, suppose that and are on the same path of DPC (the proof of and on different paths of DPC is similar to those on the same path). Let and be the endpoints of , where and . We may assume that and lie on sequentially. So there exists a neighbor (resp. ) of (resp. ) from to . Thus, the path can be separated into three vertexdisjoint sections and , where is from to , is from to and is from to . Therefore, and together with paths in DPC except form a DPC of from to . We may assume that . Let such that is an ()dimensional neighbor of for each and and are ()dimensional neighbors of and , respectively.
It is not hard to see that .
If , analogous to Condition (1), then we choose white vertices from to generate a set . By the induction hypothesis, we can obtain DPC of from to . Note that if . Let and let such that is a neighbor of for each . Clearly, . Additionally, . By the induction hypothesis, we can obtain DPC of from to . Thus, we can obtain DPC of from to (see Fig. 2).
If , analogous to Condition (2), then we can obtain DPC of from to , where contains white endpoints of DPC of . Let be a mirroring set of such that each vertex in has exactly one dimensional neighbor in , and vice versa. Clearly, and . By the induction hypothesis, we can obtain DPC of from to . Thus, we can obtain DPC of from to .
Case 1.2. . Choose white vertices from to generate a set . By regarding as , and as , in Condition (2), the proof is quite analogous to that of Case 1.1.
Case 2. and for exactly one . By symmetry of , it suffices to consider that or 2. We distinguish the following two cases.
Case 2.1. . If , then the proof is analogous to that of Case 1. So we assume that , that is, . By Lemma 6, we have . So . By the induction hypothesis, there exists DPC of from to . In addition, recall that , combining with , then we have . By adopting the same argument in Case 1.1, we can obtain DPC of from to , where contains white endpoints of DPC. Note that if . Let be a mirroring set of such that each vertex of is an ()dimensional neighbor of a vertex in . We can obtain DPC of from to .
What we have already shown is that there exists a DPC of . In what follows, we shall make some changes to DPC of , yielding a DPC of .
Suppose that is an edge on one of a path, say , of DPC of such that and lie sequentially on , where and are the endpoints of . Note that and may or may not belong to . Let and be dimensional neighbors of and , respectively. If and , then (resp. ) is an internal vertex of a path (resp. ) of DPC (resp. DPC) of (resp. ). Let (resp. ) be a neighbour of (resp. ) on (resp. ). Suppose without loss of generality that and (resp. and ) lie sequentially on (resp. ), where and (resp. and ) are the endpoints of (resp. ). Let and be dimensional neighbors of and , respectively. By Lemma 4, there exists a Hamiltonian path of from to . Deleting , and from , and and joining , , and will lead to three new paths , and , where is from to via , is from to via , and is from to via , and . By deleting edges of , and from DPC of , and adding edges of , and , a DPC of follows (see Fig. 3).
Finally, we claim that there exists an edge such that (resp. ) is not an endpoint of a path of DPC (resp. DPC) of (resp. ). Since neither nor is an endpoint of a path in DPC of , combing with the direction of on , there are at least choices of in DPC of . On the contrary, if both (resp. ) and its backup vertex are endpoints of two paths in DPC (resp. DPC) of (resp. ), then it will eliminate one choice of . Observe that and whenever . Thus, the claim holds.
Case 2.2. . Analogous to the proof of Case 1.1, we can obtain a DPC of .
Similarly, we can choose an appropriate edge on one of a path, say , of DPC of such that (resp. ) is an internal vertex of a path (resp. ) of DPC (resp. DPC) in (resp. ), where and are dimensional neighbors of and , respectively. We may assume that and lie sequentially on , where and are the endpoints of . The proof of the existence of satisfying our requirements is analogous to that of Case 2.1.
Let (resp. ) be a neighbour of (resp. ) on (resp. ). Suppose without loss of generality that and (resp. and ) lie sequentially on (resp. ), where and (resp. and ) are the endpoints of (resp. ). Let and be dimensional neighbors of and , respectively. By Lemma 4, there exists a Hamiltonian path of from to . Deleting , and from , and and joining , , and will lead to three new paths , and , where is from to via , is from to via , and , and is from to via . By deleting edges of , and from DPC of , and adding edges of , and , a DPC of follows (see Fig. 4).
Case 3. and and and for . By the relative positions of and , there are two cases to consider.
Case 3.1. or . Suppose without loss of generality that . There are two essentially distinct cases to consider.
Case 3.1.1. and . That is, and . Note that and . Additionally, and . By the proof of Case 1.1, we can obtain DPC of . Similarly, we shall make some changes to DPC of , yielding a DPC of .
Suppose that
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