Path Embeddings with Prescribed Edge in the Balanced Hypercube Network

: The balanced hypercube network, which is a novel interconnection network for parallel computation and data processing, is a newly-invented variant of the hypercube. The particular feature of the balanced hypercube is that each processor has its own backup processor and they are connected to the same neighbors. A Hamiltonian bipartite graph with bipartition V 0 ∪ V 1 is Hamiltonian laceable if there exists a path between any two vertices x ∈ V 0 and y ∈ V 1 . It is known that each edge is on a Hamiltonian cycle of the balanced hypercube. In this paper, we prove that, for an arbitrary edge e in the balanced hypercube, there exists a Hamiltonian path between any two vertices x and y in different partite sets passing through e with e (cid:54) = xy . This result improves some known results.


Introduction
Interconnection networks play an essential role in the performance of parallel and distributed systems.In the event of practice, large multi-processor systems can also be adopted as tools to address complex management and big data problems.It is well-known that an interconnection network is generally modeled by an undirected graph, in which processors are represented by vertices and communication links between them are represented by edges.The hypercube network is recognized as one of the most popular interconnection networks, and it has gained great attention and recognition from researchers both in graph theory and computer science.Nevertheless, the hypercube also has some shortcomings.For example, its diameter is large.Therefore, many variants of the hypercube have been put forward [1][2][3][4][5][6][7][8][9][10] to improve performance of the hypercube in some aspects.Among these variants, the balanced hypercube has the following special properties: each vertex of the balanced has a backup (matching) vertex and they have the same neighborhood.Therefore, the backup vertex can undertake tasks that originally run on a faulty vertex.It has been proved that the diameter of an odd-dimensional balanced hypercube BH n is 2n − 1 [10], which is smaller than that of the hypercube Q 2n .
With regard to the special properties discussed above, the balanced hypercube has been investigated by many researchers.Huang and Wu [11] studied the problem of resource placement of the balanced hypercube.Xu et al. [12] showed that the balanced hypercube is edge-pancyclic and Hamiltonian laceable.It is found that the balanced hypercube is bipanconnected for all n ≥ 1 by Yang [13].Huang et al. [14] discussed area efficient layout problems of the balanced hypercube.Yang [15] determined super (edge) connectivity of the balanced hypercube.Lü et al. studied (conditional) matching preclusion, hyper-Hamiltonian laceability, matching extendability and extra connectivity of the balanced hypercube in [16][17][18][19], respectively.Some symmetric properties of the balanced hypercube are presented in [20,21].As stated above, the balanced hypercube possesses some desirable properties that the hypercube does not have, so it is interesting to explore other favorable properties that the balanced hypercube may have.
Since parallel applications such as image and signal processing are originally designed on array and ring architectures, it is important to have path and cycle embeddings in a network.Especially, Hamiltonian path and cycle embeddings and other properties of famous networks are extensively studied by many authors [12,13,[22][23][24][25][26].Xu et al. [12] proved that each edge of the balanced hypercube is on a cycle of even length from 4 to 4 n , that is, the balanced hypercube is edge-bipancyclic.They also showed that the balanced hypercube is Hamiltonian laceable for all integers n ≥ 1.Recently, Lü et al. [17] further obtained that the balanced hypercube is hyper-Hamiltonian laceable for all integers n ≥ 1.
The rest of this paper is organized as follows.Some necessary definitions are presented as preliminaries in Section 2. The main result of this paper is shown in Section 3. Finally, conclusions are given in Section 4.

Preliminaries
Let G = (V, E) be a simple undirected graph, where V is a vertex-set of G and E is an edge-set of G.A path P from v 0 to v n is a sequence of vertices v 0 v 1 • • • v n from v 0 to v n such that every pair of consecutive vertices are adjacent and all vertices are distinct except for v 0 and v n .We also denote the path The length of a path P is the number of edges in P, denoted by l(P).A cycle is a path with at least three vertices such that the first vertex is the same as the last one.A graph is bipartite if its vertex-set can be partitioned into two subsets V 0 and V 1 such that each edge has its ends in different subsets.A graph is Hamiltonian if it possesses a spanning cycle.A graph is Hamiltonian connected if there exists a Hamiltonian path joining any two vertices of it.Obviously, any bipartite graph is not Hamiltonian connected.Simmons [27] proposed Hamiltonian laceability of bipatite graphs: a bipartite graph G = (V 0 ∪ V 1 , E) is Hamiltonian laceable if there exists a Hamiltonian path between any two vertices x and y in different partite sets of G.A graph G is hyper-Hamiltonian laceable if it is Hamiltonian laceable and, for any vertex v ∈ V i (i ∈ {0, 1}), there exists a Hamiltonian path in G − v between any pair of vertices in V 1−i .For the graph definitions and notations not mentioned here, we refer the readers to [28,29].
Wu and Huang [10] gave the following definition of BH n as follows.
In BH n , the first coordinate a 0 of vertex (a 0 , . . ., a i , . . ., a n−1 ) is called the inner index and the other coordinates are known as the a i (1 ≤ i ≤ n − 1) i-dimensional index.Clearly, each vertex in BH n has two inner neighbors, and 2n − 2 other neighbors.Note that all of the arithmetic operations on indices of vertices in BH n are four-modulated.
BH 1 and BH 2 are illustrated in Figures 1 and 2, respectively.( ) 0,0 In the following, we give some basic properties of BH n .

Main Results
Firstly, we characterize edges of the BH n .Let u and v be two adjacent vertices in BH n .If u and v differ in only the inner index, then uv is said to be a 0-dimensional edge, and u is a 0-dimensional neighbor of v.If u and v differ in not only the inner index, but also some i-dimensional index (i = 0) of the vertices, then uv is called an i-dimensional edge, and u is an i-dimensional neighbor of v.For convenience, we denote the set of all i-dimensional edges by ∂D i (0 (0 ≤ i ≤ 3) be the subgraph of BH n induced by the vertices of BH n with the (n − 1)-dimensional index i.That is, the BH (i) n−1 's can be obtained from BH n by deleting all (n − 1)-dimensional edges.Therefore, BH By Proposition 1, we know that BH n is bipartite.We can use V 0 and V 1 to denote the two partite sets of BH n such that V 0 and V 1 consist of vertices of BH n with an even inner index and an odd inner index, respectively.For convenience, the vertices of V 0 and V 1 are colored white and black, respectively.Throughout this paper, we use w i and u i (resp.b i and v i ) to denote white (resp.black) vertices in Lemma 1. [16] In BH n , ∂D i (0 ≤ i ≤ n − 1) can be divided into 4 n−1 edge-disjoint 4-cycles for n ≥ 1. Lemma 2. [12] The balanced hypercube BH n is Hamiltonian laceable and edge-bipancyclic for n ≥ 1. Lemma 3. [17] The balanced hypercube BH n is hyper-Hamiltonian laceable for n ≥ 1. Lemma 4. [30] Assume u and x are two different vertices in V 0 , and v and y are two different vertices in V 1 .Then, there exist two vertex-disjoint paths P and Q such that P joins x to y, Q joins u to v and V(P) ∪ V(Q) = V(BH n ), where n ≥ 1.

Lemma 5.
Let n ≥ 2 be an integer.Suppose that u, v, x and y are four distinct vertices differ only the inner index in BH n .In addition, u, x ∈ V 0 and v, y ∈ V 1 .Then, there exists a Hamiltonian path from u to v in BH n − x − y.
Proof.We proceed with the proof by the induction on n.First, we consider n = 2. Clearly, u, v, x and y are in the same 4-cycle of ∂D 0 .A Hamiltonian path of BH 2 − x − y from u to v is shown in Figure 3. Thus, we suppose that the lemma holds for all integers n − 1 with n ≥ 3. Next, we consider BH n .We split BH n into four BH n−1 s by deleting (n − 1)-dimensional edges.For convenience, we denote the four BH n−1 s by B 0 , B 1 , B 2 and B 3 according to the last position of vertices in BH n , respectively.Without loss of generality, we may assume that u, v, x and y are in B 0 .By an induction hypothesis, there exists a Hamiltonian path P 0 from u to v in B 0 − x − y.Let u 0 v 0 ∈ E(P 1 ), where u 0 (resp.v 0 ) are neither end-vertex of P 0 .We denote the segment of P 0 from u to v 0 by P 00 , and the segment of P 0 from u 0 to v by P 10 .By Definition 1, u 0 (resp.v 0 ) has an (n − 1)-dimensional neighbor v 1 (resp.u 3 ) in B 1 (resp.B 3 ).Moreover, there exist an edge v 3 u 2 from B 3 to B 2 , and an edge v 2 u 1 from B 2 to B 1 .Therefore, there exist a Hamiltonian path P 3 from u 3 to v 3 in B 3 , a Hamiltonian path P 2 from u 2 to v 2 in B 2 , and a Hamiltonian path P 1 from u 1 to v 1 of B 1 .Hence, u, P 00 , v 0 , u 3 , P 3 , v 3 , u 2 , P 2 , v 2 , u 1 , P 1 , v 1 , u 0 , P 10 , v is a Hamiltonian path of BH n − x − y (see Figure 4).Next, we present the following lemma as a basis of our main theorem.Lemma 6.Let e be an arbitrary edge in BH 2 .In addition, let x ∈ V 1 and y ∈ V 0 be any two vertices in BH 2 with e = xy.Then, there exists a Hamiltonian path between x and y passing through e.
Proof.By Proposition 2, BH 2 is vertex-transitive and edge-transitive, and we may suppose that e = (0, 0)(1, 0).Obviously, if e = xy, then there exists no Hamiltonian path of BH 2 from x to y passing e.Thus, at most, one of x and y is the end-vertex of e.We consider the following two cases: Case 1: Neither x nor y is incident to e.By the relative positions of x and y, and Proposition 3, we consider the following: (1) For simplicity, we list all Hamiltonian paths of the conditions above in Table 1.
Case 2: Either x or y is incident to e. Without loss of generality, suppose that x is incident to e, that is, x = (1, 0).By Proposition 3, we need only to consider four conditions of y: (1) y ∈ V(B 0 ); (2) y ∈ V(B 1 ); (3) y ∈ V(B 2 ); and (4) y ∈ V(B 3 ).Again, we list Hamiltonian paths of the conditions of x and y in this case in Table 2.
Now, we are ready to state the main theorem of this paper.
Theorem 1.Let n ≥ 2 be an integer and e be an arbitrary edge in BH n .In addition, let x ∈ V 1 and y ∈ V 0 be any two vertices in BH n with e = xy.Then, there exists a Hamiltonian path of BH n between x and y passing through e.
Proof.We prove this theorem by induction on n.By Lemma 6, we know that the theorem is true for n = 2. Therefore, we suppose that the theorem holds for n − 1 with n ≥ 3. Next, we consider BH n .Firstly, we divide BH n into BH n−1 (0 ≤ i ≤ 3) by deleting all (n − 1)-dimensional edges.For convenience, we denote BH (i) n−1 by B i according to the last position of the vertices in BH n for each i ∈ {0, 1, 2, 3}.Similarly, suppose that e ∈ E(B 0 ).Let x ∈ V 1 and y ∈ V 0 be two distinct vertices in BH n .By relative positions of x and y, we consider the following cases: . By an induction hypothesis, there exists a Hamiltonian path P 0 from x to y of B 0 passing through e.Thus, there is an edge u 0 v 0 on P 0 such that u 0 v 0 is not adjacent to e and u 0 v 0 divides P 0 into two sections P 00 and P 10 , where P 00 connects x to u 0 and P 10 connects v 0 to y.Let v 1 (resp.u 3 ) be an (n − 1)-dimensional neighbor of u 0 (resp.v 0 ).By Definition 1, there exist an edge u 1 v 2 from B 1 to B 2 , and an edge u 2 v 3 from B 2 to B 3 .Thus, by Lemma 2, there exist a Hamiltonian path P 1 from v 1 to u 1 in B 1 , a Hamiltonian path P 2 from v 2 to u 2 in B 2 , and a Hamiltonian path P 3 from v 3 to u 3 in B 3 .Hence, x, P 00 , u 0 , v 1 , P 1 , u 1 , v 2 , P 2 , u 2 , v 3 , P 3 , u 3 , v 0 , P 10 , y is a Hamiltonian path of BH n from x to y passing through e (see Figure 5).Case 2: x ∈ V(B 0 ), y ∈ V(B 1 ).Let u 0 ∈ V(B 0 ) be a white vertex such that u 0 is not incident to e.By an induction hypothesis, there exists a Hamiltonian path P 0 of B 0 from x to u 0 passing through e. Supposing that v 0 is a black vertex adjacent to u 0 on P 0 , we denote the segment of the path P 0 from x to v 0 by P 00 .Let the two (n − 1)-dimensional neighbors of u 0 be b 1 and v 1 .By Lemma 2, there exists a Hamiltonian path P 1 of B 1 from b 1 to y.Let u 1 be the neighbor of v 1 in the section of P 1 from b 1 to v 1 .Then P 1 − u 1 v 1 consists of two subpaths P 01 and P 11 , which connect u 1 to b 1 and v 1 to y, respectively.
through e with Neither x nor y Being Incident to e

Table 1 .
Hamiltonian paths passing through e with neither x nor y being incident to e.

Table 2 .
Hamiltonian paths passing through e with x or y being incident to e.