S S smmetr Communication Path Embeddings with Prescribed Edge in the Balanced Hpercube Network Dan Chen, Zhongzhou Lu, Zebang Shen, Gaofeng Zhang, Chong Chen and Qingguo Zhou * School of Information Science and Engineering, Lanzhou Universit, Lanzhou, Gansu 730000, China; chend13@lzu.edu.cn (D.C.); luzhzh15@lzu.edu.cn (Z.L.); shenzb12@lzu.edu.cn (Z.S.); zhanggaof@lzu.edu.cn (G.Z.); chench2013@lzu.edu.cn (C.C.) * Correspondence: zhouqg@lzu.edu.cn Academic Editor: Angel Garrido Received: 21 Februar 2017; Accepted: 15 Ma 2017; Published: 26 Ma 2017 Abstract: The balanced hpercube network, which is a novel interconnection network for parallel computation and data processing, is a newl-invented variant of the hpercube. The particular feature of the balanced hpercube is that each processor has its own backup processor and the are connected to the same neighbors. A Hamiltonian bipartite graph with bipartition V 0 V 1 is Hamiltonian laceable if there eists a path between an two vertices V 0 and V 1. It is known that each edge is on a Hamiltonian ccle of the balanced hpercube. In this paper, we prove that, for an arbitrar edge e in the balanced hpercube, there eists a Hamiltonian path between an two vertices and in different partite sets passing through e with e =. This result improves some known results. Kewords: interconnection network; balanced hpercube; Hamiltonian path; passing prescribed edge; data processing 1. Introduction Interconnection networks pla an essential role in the performance of parallel and distributed sstems. In the event of practice, large multi-processor sstems can also be adopted as tools to address comple management and big data problems. It is well-known that an interconnection network is generall modeled b an undirected graph, in which processors are represented b vertices and communication links between them are represented b edges. The hpercube network is recognized as one of the most popular interconnection networks, and it has gained great attention and recognition from researchers both in graph theor and computer science. Nevertheless, the hpercube also has some shortcomings. For eample, its diameter is large. Therefore, man variants of the hpercube have been put forward [1 10] to improve performance of the hpercube in some aspects. Among these variants, the balanced hpercube has the following special properties: each verte of the balanced has a backup (matching) verte and the have the same neighborhood. Therefore, the backup verte can undertake tasks that originall run on a fault verte. It has been proved that the diameter of an odd-dimensional balanced hpercube BH n is 2n 1 [10], which is smaller than that of the hpercube Q 2n. With regard to the special properties discussed above, the balanced hpercube has been investigated b man researchers. Huang and Wu [11] studied the problem of resource placement of the balanced hpercube. Xu et al. [12] showed that the balanced hpercube is edge-pancclic and Hamiltonian laceable. It is found that the balanced hpercube is bipanconnected for all n 1 b Yang [13]. Huang et al. [14] discussed area efficient laout problems of the balanced hpercube. Yang [15] determined super (edge) connectivit of the balanced hpercube. Lü et al. studied (conditional) matching preclusion, hper-hamiltonian laceabilit, matching etendabilit and etra connectivit of the balanced hpercube in [16 19], respectivel. Some smmetric properties of the Smmetr 2017, 9, 79; doi:10.3390/sm9060079 www.mdpi.com/journal/smmetr
Smmetr 2017, 9, 79 2 of 12 balanced hpercube are presented in [20,21]. As stated above, the balanced hpercube possesses some desirable properties that the hpercube does not have, so it is interesting to eplore other favorable properties that the balanced hpercube ma have. Since parallel applications such as image and signal processing are originall designed on arra and ring architectures, it is important to have path and ccle embeddings in a network. Especiall, Hamiltonian path and ccle embeddings and other properties of famous networks are etensivel studied b man authors [12,13,22 26]. Xu et al. [12] proved that each edge of the balanced hpercube is on a ccle of even length from 4 to 4 n, that is, the balanced hpercube is edge-bipancclic. The also showed that the balanced hpercube is Hamiltonian laceable for all integers n 1. Recentl, Lü et al. [17] further obtained that the balanced hpercube is hper-hamiltonian laceable for all integers n 1. The rest of this paper is organized as follows. Some necessar definitions are presented as preliminaries in Section 2. The main result of this paper is shown in Section 3. Finall, conclusions are given in Section 4. 2. Preliminaries Let G = (V, E) be a simple undirected graph, where V is a verte-set of G and E is an edge-set of G. A path P from to v n is a sequence of vertices v n from to v n such that ever pair of consecutive vertices are adjacent and all vertices are distinct ecept for and v n. We also denote the path P = v n b, P, v n. The length of a path P is the number of edges in P, denoted b l(p). A ccle is a path with at least three vertices such that the first verte is the same as the last one. A graph is bipartite if its verte-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 ccle. A graph is Hamiltonian connected if there eists a Hamiltonian path joining an two vertices of it. Obviousl, an bipartite graph is not Hamiltonian connected. Simmons [27] proposed Hamiltonian laceabilit of bipatite graphs: a bipartite graph G = (V 0 V 1, E) is Hamiltonian laceable if there eists a Hamiltonian path between an two vertices and in different partite sets of G. A graph G is hper-hamiltonian laceable if it is Hamiltonian laceable and, for an verte v V i (i {0, 1}), there eists a Hamiltonian path in G v between an 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. Definition 1. An n-dimensional balanced hpercube, denoted b BH n, consists of 4 n vertices labelled b (a 0, a 1,..., a n 1 ), where a i {0, 1, 2, 3} for each 0 i n 1. An verte (a 0,..., a i 1, a i, a i+1,..., a n 1 ) with 1 i n 1 of BH n has the following 2n neighbors: 1. ((a 0 + 1) mod 4, a 1,..., a i 1, a i, a i+1,..., a n 1 ), ((a 0 1) mod 4, a 1,..., a i 1, a i, a i+1,..., a n 1 ), and 2. ((a 0 + 1) mod 4, a 1,..., a i 1, (a i + ( 1) a 0) mod 4, a i+1,..., a n 1 ), ((a 0 1) mod 4, a 1,..., a i 1, (a i + ( 1) a 0) mod 4, a i+1,..., a n 1 ).
Smmetr 2017, 9, 79 3 of 12 In BH n, the first coordinate a 0 of verte (a 0,..., a i,..., a n 1 ) is called the inner inde and the other coordinates are known as the a i (1 i n 1) i-dimensional inde. Clearl, each verte 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, respectivel. 0 1 3 2 Figure 1. BH 1. ( 0,0) ( 1,0) ( 0,1) ( 1,1) ( 3,0) ( 2,0) ( 3,1) ( 2,1) ( 0,3) ( 1,3) ( 0,2) ( 1,2) ( 3,3) ( 2,3) ( 3,2) ( 2,2) Figure 2. BH 2. In the following, we give some basic properties of BH n. Proposition 1. [10] The balanced hpercube is bipartite. Proposition 2. [10,20] The balanced hpercube is verte-transitive and edge-transitive. Proposition 3. [10] The vertices (a 0, a 1,..., a n 1 ) and ((a 0 + 2) mod 4, a 1,..., a n 1 ) of BH n have the same neighborhood. 3. Main Results Firstl, we characterize edges of the BH n. Let u and v be two adjacent vertices in BH n. If u and v differ in onl the inner inde, 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 onl the inner inde, but also some i-dimensional inde (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 b D i (0 i n 1). Let BH (i) n 1 (0 i 3) be the subgraph of BH n induced b the vertices of BH n with the (n 1)-dimensional
Smmetr 2017, 9, 79 4 of 12 inde i. That is, the BH (i) n 1 s can be obtained from BH n b deleting all (n 1)-dimensional edges. Therefore, BH (i) n 1 = BH n 1 for each 0 i 3. B 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 inde and an odd inner inde, respectivel. For convenience, the vertices of V 0 and V 1 are colored white and black, respectivel. Throughout this paper, we use w i and u i (resp. b i and v i ) to denote white (resp. black) vertices in BH (i) n 1 (i {0, 1, 2, 3}). Lemma 1. [16] In BH n, D i (0 i n 1) can be divided into 4 n 1 edge-disjoint 4-ccles for n 1. Lemma 2. [12] The balanced hpercube BH n is Hamiltonian laceable and edge-bipancclic for n 1. Lemma 3. [17] The balanced hpercube BH n is hper-hamiltonian laceable for n 1. Lemma 4. [30] Assume u and are two different vertices in V 0, and v and are two different vertices in V 1. Then, there eist two verte-disjoint paths P and Q such that P joins to, 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, and are four distinct vertices differ onl the inner inde in BH n. In addition, u, V 0 and v, V 1. Then, there eists a Hamiltonian path from u to v in BH n. Proof. We proceed with the proof b the induction on n. First, we consider n = 2. Clearl, u, v, and are in the same 4-ccle of D 0. A Hamiltonian path of BH 2 from u to v is shown in Figure 3. Thus, we suppose that the lemma holds for all integers n 1 with n 3. Net, we consider BH n. We split BH n into four BH n 1 s b deleting (n 1)-dimensional edges. For convenience, we denote the four BH n 1 s b B 0, B 1, and according to the last position of vertices in BH n, respectivel. Without loss of generalit, we ma assume that u, v, and are in B 0. B an induction hpothesis, there eists a Hamiltonian path P 0 from u to v in B 0. Let E(P 1 ), where (resp. ) are neither end-verte of P 0. We denote the segment of P 0 from u to b P 00, and the segment of P 0 from to v b P 10. B Definition 1, (resp. ) has an (n 1)-dimensional neighbor (resp. u 3 ) in B 1 (resp. ). Moreover, there eist an edge from to, and an edge from to B 1. Therefore, there eist a Hamiltonian path from u 3 to in, a Hamiltonian path P 2 from to in, and a Hamiltonian path P 1 from to of B 1. Hence, u, P 00,, u 3,,,, P 2,,, P 1,,, P 10, v is a Hamiltonian path of BH n (see Figure 4). u v Figure 3. A Hamiltonian path of BH 2.
Smmetr 2017, 9, 79 5 of 12 v u v P 1 P 10 1 P 00 u3 v3 P 2 Figure 4. A Hamiltonian path of BH n. Net, we present the following lemma as a basis of our main theorem. Lemma 6. Let e be an arbitrar edge in BH 2. In addition, let V 1 and V 0 be an two vertices in BH 2 with e =. Then, there eists a Hamiltonian path between and passing through e. Proof. B Proposition 2, BH 2 is verte-transitive and edge-transitive, and we ma suppose that e = (0, 0)(1, 0). Obviousl, if e =, then there eists no Hamiltonian path of BH 2 from to passing e. Thus, at most, one of and is the end-verte of e. We consider the following two cases: Case 1: Neither nor is incident to e. B the relative positions of and, and Proposition 3, we consider the following: (1) V(B 0 ), V(B 0 ); (2) V(B 0 ), V(B 1 ); (3) V(B 0 ), V( ); (4) V(B 0 ), V( ); (5) V(B 1 ), V(B 1 ); (6) V(B 1 ), V( ); (7) V(B 1 ), V( ); (8) V( ), V( ); (9) V( ), V( ); (10) V( ), V( ). For simplicit, we list all Hamiltonian paths of the conditions above in Table 1. Case 2: Either or is incident to e. Without loss of generalit, suppose that is incident to e, that is, = (1, 0). B Proposition 3, we need onl to consider four conditions of : (1) V(B 0 ); (2) V(B 1 ); (3) V( ); and (4) V( ). Again, we list Hamiltonian paths of the conditions of and in this case in Table 2. Table 1. Hamiltonian paths passing through e with neither nor being incident to e. Hamiltonian Paths Passing through e with Neither nor Being Incident to e (1) (3,0) (2,0) (3,0)(0,3)(3,3)(2,3)(1,3)(0,2)(3,2)(2,2)(1,2)(2,1)(3,1)(0,1)(1,1)(0,0)(1,0)(2,0) (2) (3,0) (0,1) (3,0)(0,0)(1,0)(2,3)(3,3)(0,3)(1,3)(0,2)(3,2)(2,2)(1,2)(2,1)(3,1)(2,0)(1,1)(0,1) (3) (3,0) (2,2) (3,0)(0,3)(3,3)(2,3)(1,0)(0,0)(3,1)(2,0)(1,1)(0,1)(1,2)(2,1)(3,2)(0,2)(1,3)(2,2) (4) (3,0) (0,3) (3,0)(0,0)(1,0)(2,0)(3,1)(0,1)(1,1)(2,1)(1,2)(2,2)(3,2)(0,2)(1,3)(2,3)(3,3)(0,3) (5) (1,1) (2,1) (1,1)(0,1)(3,1)(2,0)(1,0)(0,0)(3,0)(0,3)(3,3)(2,3)(1,3)(0,2)(3,2)(2,2)(1,2)(2,1) (6) (1,1) (2,2) (1,1)(0,1)(3,1)(2,0)(1,0)(0,0)(3,0)(0,3)(3,3)(2,3)(1,3)(0,2)(3,2)(2,1)(1,2)(2,2) (7) (1,1) (2,3) (1,1)(0,0)(3,1)(0,1)(1,2)(2,1)(3,2)(2,2)(1,3)(0,2)(3,3)(0,3)(1,0)(2,0)(3,0)(2,3) (8) (1,2) (2,2) (1,2)(2,1)(1,1)(0,1)(3,1)(2,0)(1,0)(0,0)(3,0)(0,3)(3,3)(2,3)(1,3)(0,2)(3,2)(2,2) (9) (1,2) (2,3) (1,2)(2,1)(1,1)(0,1)(3,1)(2,0)(1,0)(0,0)(3,0)(0,3)(1,3)(2,2)(3,2)(0,2)(3,3)(2,3) (10) (1,3) (2,3) (1,3)(0,3)(3,0)(0,0)(1,0)(2,0)(1,1)(2,1)(3,1)(0,1)(3,2)(2,2)(1,2)(0,2)(3,3)(2,3)
Smmetr 2017, 9, 79 6 of 12 Table 2. Hamiltonian paths passing through e with or being incident to e. Hamiltonian Paths Passing through e with or Being Incident to e (1) (1,0) (2,0) (1,0)(0,0)(3,0)(0,3)(3,3)(2,3)(1,3)(0,2)(3,2)(2,2)(1,2)(2,1)(1,1)(0,1)(3,1)(2,0) (2) (1,0) (0,1) (1,0)(0,0)(3,0)(0,3)(3,3)(2,3)(1,3)(0,2)(3,2)(2,2)(1,2)(2,1)(1,1)(2,0)(3,1)(0,1) (3) (1,0) (0,2) (1,0)(0,0)(3,0)(0,3)(1,3)(2,3)(3,3)(2,2)(3,2)(2,1)(1,1)(2,0)(3,1)(0,1)(1,2)(0,2) (4) (1,0) (0,3) (1,0)(0,0)(3,0)(2,0)(3,1)(0,1)(1,1)(2,1)(1,2)(2,2)(3,2)(0,2)(1,3)(2,3)(3,3)(0,3) Now, we are read to state the main theorem of this paper. Theorem 1. Let n 2 be an integer and e be an arbitrar edge in BH n. In addition, let V 1 and V 0 be an two vertices in BH n with e =. Then, there eists a Hamiltonian path of BH n between and passing through e. Proof. We prove this theorem b induction on n. B 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. Net, we consider BH n. Firstl, we divide BH n into BH (i) n 1 (0 i 3) b deleting all (n 1)-dimensional edges. For convenience, we denote BH (i) n 1 b B i according to the last position of the vertices in BH n for each i {0, 1, 2, 3}. Similarl, suppose that e E(B 0 ). Let V 1 and V 0 be two distinct vertices in BH n. B relative positions of and, we consider the following cases: Case 1: V(B 0 ), V(B 0 ). B an induction hpothesis, there eists a Hamiltonian path P 0 from to of B 0 passing through e. Thus, there is an edge on P 0 such that is not adjacent to e and divides P 0 into two sections P 00 and P 10, where P 00 connects to and P 10 connects to. Let (resp. u 3 ) be an (n 1)-dimensional neighbor of (resp. ). B Definition 1, there eist an edge from B 1 to, and an edge from to. Thus, b Lemma 2, there eist a Hamiltonian path P 1 from to in B 1, a Hamiltonian path P 2 from to in, and a Hamiltonian path from to u 3 in. Hence,, P 00,,, P 1,,, P 2,,,, u 3,, P 10, is a Hamiltonian path of BH n from to passing through e (see Figure 5). P 00 P 10 P 1 u3 v3 P 2 Figure 5. Illustration for Case 1. Case 2: V(B 0 ), V(B 1 ). Let V(B 0 ) be a white verte such that is not incident to e. B an induction hpothesis, there eists a Hamiltonian path P 0 of B 0 from to passing through e. Supposing that is a black verte adjacent to on P 0, we denote the segment of the path P 0 from to b P 00. Let the two (n 1)-dimensional neighbors of be b 1 and. B Lemma 2, there eists a Hamiltonian path P 1 of B 1 from b 1 to. Let be the neighbor of in the section of P 1 from b 1 to. Then P 1 consists of two subpaths P 01 and P 11, which connect to b 1 and to, respectivel.
Smmetr 2017, 9, 79 7 of 12 Let u 3 (resp. ) be an (n 1)-dimensional neighbor of (resp. ). Furthermore, there eists an edge from to. Then, there eist a Hamiltonian path P 2 from to in, and a Hamiltonian path from u 3 to in. Hence,, P 00,, u 3,,,, P 2,,, P 01, b 1,,, P 11, is a Hamiltonian path of BH n from to passing through e (see Figure 6). P 00 b1 P 01 P 11 u3 v3 P 2 Figure 6. Illustration for Case 2. Case 3: V(B 0 ), V( ). Let be a white verte in B 0 not incident to e, and b 1 and be two (n 1)-dimensional neighbors of. In addition, assume that w 1 is an arbitrar white verte in B 1. There eists a Hamiltonian path of B 1 from b 1 to w 1. Thus, there eists an edge E(P 1 ) whose removal will lead to two disjoint subpaths P 01 and P 11, where P 01 connects to b 1 and P 11 connects to w 1. Let (resp. b 2 ) be an (n 1)-dimensional neighbor of (resp. w 1 ). There also eists a Hamiltonian path P 2 of from to b 2 via the edge. Deleting results in two disjoint paths P 02 and P 12, where P 02 connects to b 2 and P 12 connects to. B an induction hpothesis, there eists a Hamiltonian path P 0 of B 0 from to via the edge. For convenience, denote P 0 b P 00, that is, P 00 connects to. Let u 3 (resp. ) be an (n 1)-dimensional neighbor of (resp. ). Again, there eists a Hamiltonian path of from u 3 to. Hence,, P 00,, u 3,,,, P 02, b 2, w 1, P 11,,, b 1, P 01,,, P 12, is a Hamiltonian path of BH n from to passing through e (see Figure 7). u 3 P 00 b1 P 01 P 12 P 02 P 11 w 1 b 2 Figure 7. Illustration for Case 3. Case 4: V(B 0 ), V( ). Let (resp. ) be a white (resp. black) verte in B 0 (resp. ). There eist an edge from B 0 to B 1, an edge from B 1 to, and an edge from to.
Smmetr 2017, 9, 79 8 of 12 B Lemma 2, there eist a Hamiltonian path P 1 of B 1 from to, a Hamiltonian path P 2 of from to, and a Hamiltonian path of from to u 3. B an induction hpothesis, there eists a Hamiltonian path P 0 of B 0 from to passing through e. Hence,, P 0,,, P 1,,, P 2,,,, is a Hamiltonian path of BH n from to passing through e (see Figure 8). P 0 u v 0 1 P1 P 2 Figure 8. Illustration for Case 4. Case 5: V(B 1 ), V(B 1 ). Let = be a black verte in B 1. B Lemma 3, there eists a Hamiltonian path P 1 of B 1 from to. Furthermore, there eist an edge from B 1 to B 0, an edge u 3 from B 0 to, an edge from to, and an edge from to B 1. Moreover, there eist a Hamiltonian path P 0 of B 0 from to passing through e, a Hamiltonian path of from u 3 to, and a Hamiltonian path P 2 of from to. Hence,, P 1,,, P 0,, u 3,,,, P 2,, is a Hamiltonian path of BH n from to passing through e (see Figure 9). P 0 P 1 v2 u3 P P 2 3 Figure 9. Illustration for Case 5. Case 6: V(B 1 ), V( ). Let = (resp. ) be a black (resp. white) verte in B 1. B Lemma 3, there eists a Hamiltonian path P 1 of B 1 from to. In addition, suppose that and b 2 are two (n 1)-dimensional neighbors of. B Lemma 2, there eists a Hamiltonian P 2 of from to via the edge b 2. Thus, P 2 can be divided into three sections: P 02, and P 12, where P 02 connects to and P 12 connects b 2 to. Furthermore, there eist an edge from B 1 to B 0, an edge u 3 from B 0 to, and an edge from to. Therefore, there eist a Hamiltonian path P 0 of B 0 from to passing through e, and a Hamiltonian path of from u 3 to. Hence,, P 1,,, P 0,, u 3,,,, P 02,,, b 2, P 12, is a Hamiltonian path of BH n from to passing through e (see Figure 10).
Smmetr 2017, 9, 79 9 of 12 P 0 P u0 1 P 02 b 2 u 3 P 12 Figure 10. Illustration for Case 6. Case 7: V(B 1 ), V( ). Let and b 3 be two black vertices in. Suppose that and w 2 are (n 1)-dimensional neighbors of and b 2, respectivel. B Lemma 3, there eists a Hamiltonian path of from b 3 to. B Definition 1, there eist two edges and b 2 w 1 from to B 1, an edge from B 1 to B 0, and an edge from B 0 to, where =. B Lemma 4, there eist two verte-disjoint paths P 01 and P 11 such that P 01 joins and, P 11 joins and w 1, and V(P 01 ) V(P 11 ) = V(B 1 ). Similarl, there eist two verte-disjoint paths P 02 and P 12 such that P 02 joins and, P 12 joins b 2 and w 2, and V(P 02 ) V(P 12 ) = V( ). B an induction hpothesis, there eists a Hamiltonian path P 0 of B 0 from to passing through e. Hence,, P 11, w 1, b 2, P 12, w 2, b 3,,,, P 02,,, P 01,,, P 0,, is a Hamiltonian path of BH n from to passing through e (see Figure 11). v P 11 0 P 0 P 01 w 1 P 02 b 2 b 3 w 2 P 12 Figure 11. Illustration for Case 7. Case 8: V( ), V( ). Let V( ) be an arbitrar white verte. B Lemma 3, there eists a Hamiltonian path P 2 of from to. B Definition 1, there eist an edge from to B 1, an edge from B 1 to B 0, an edge u 3 from B 0 to, and an edge from to. Following Lemma 2, we can obtain a Hamiltonian path P 1 of B 1 from to, and a Hamiltonian path of from u 3 to. B an induction hpothesis, there eists a Hamiltonian path P 0 of B 0 from to passing through e. Therefore,,, P 1,,, P 0,, u 3,,,, P 2, is a Hamiltonian path of BH n from to passing through e (see Figure 12).
Smmetr 2017, 9, 79 10 of 12 P 0 P 1 u 3 P 2 Figure 12. Illustration for Case 8. Case 9: V( ), V( ). Let and w 2 be two distinct white vertices in, and and b 3 be (n 1)-dimensional neighbors of and w 2, respectivel. B Lemma 3, there eists a Hamiltonian path P 2 of from to w 2. B Lemma 2, there eists a Hamiltonian path of from to via the edge u 3 b 3. B deleting u 3 b 3, we can obtain two disjoint subpaths: P 03 and P 13, where P 03 connects u 3 to and P 13 connects b 3 to. Furthermore, there eist an edge from to B 1, an edge from B 1 to B 0, and an edge u 3 from B 0 to. B Lemma 2, there eists a Hamiltonian path P 1 of B 1 from to. B an induction hpothesis, there eists a Hamiltonian path P 0 of B 0 from to passing through e. Hence,,, P 1,,, P 0,, u 3, P 03,,, P 2, w 2, b 3, P 13, is a Hamiltonian path of BH n from to passing through e (see Figure 13). P 0 P 1 u P 03 3 P 13 w b3 3 P 2 Figure 13. Illustration for Case 9. Case 10: V( ), V( ). The proof is analogous to that of Case 5, and we omit it. 4. Conclusions In this paper, we stud a tpe of path embedding of the balanced hpercube, and show that, for an arbitrar edge e =, there eists a Hamiltonian path between an two vertices and in different partite sets passing through e. This result also implies that each edge is on a Hamiltonian ccle of the balanced hpercube, which is part of the results of edge bipancclicit of the balanced hpercube. Acknowledgments: This work was supported b National Natural Science Foundation of China under Grant Nos. 61402210 and 60973137, Program for New Centur Ecellent Talents in Universit under Grant No. NCET-12-0250,
Smmetr 2017, 9, 79 11 of 12 Major Project of High Resolution Earth Observation Sstem with Grant No. 30-Y20A34-9010-15/17, Strategic Priorit Research Program of the Chinese Academ of Sciences with Grant No. XDA03030100, Gansu Sci.&Tech. Program under Grant Nos. 1104GKCA049, 1204GKCA061 and 1304GKCA018, The Fundamental Research Funds for the Central Universities under Grant No. lzujbk-2016-140, Gansu Telecom Cuiing Research Fund under Grant No. lzudc-2013-4, Google Research Awards and Google Facult Award, China. Author Contributions: Dan Chen and Chong Chen initiated the research idea and developed the models with contributions from Gaofeng Zhang. Zhongzhou Lu and Zebang Shen offered help while constructing the model. The manuscript was written b Dan Chen and Chong Chen with Rui Zhou and Qingguo Zhou providing review and comments. All the authors were engaged in the final manuscript preparation and agreed to the publication of this paper. Conflicts of Interest: The authors declare no conflict of interest. References 1. Abraham, S.; Padmanabhan, K. The twisted cube topolog for multiprocessors: A stud in network asmmetr. J. Parall. Distrib. Comput. 1991, 13, 104 110, doi:10.1016/0743-7315(91)90113-n. 2. Choudum, S.A.; Sunitha, V. Augmented cubes. Networks 2002, 40, 71 84. 3. Cull, P.; Larson, S.M. The Möbius cubes. IEEE Trans. Comput. 1995, 44, 647 659. 4. Dall, W.J. Performance analsis of k-ar n-cube interconnection networks. IEEE Trans. Comput. 1990, 39, 775 785. 5. Efe, K. The crossed cube architecture for parallel computation. IEEE Trans. Parall. Distr. Sst. 1992, 3, 513 524. 6. El-amaw, A.; Latifi, S. Properties and performance of folded hpercubes. IEEE Trans. Parall. Distrib. Sst. 1991, 2, 31 42. 7. Li, T.K.; Tan, J.J.M.; Hsu, L.H.; Sung, T.Y. The shuffle-cubes and their generalization. Inform. Process. Lett. 2001, 77, 35 41. 8. Preparata, F.P.; Vuillemin, J. The cube-connected ccles: A versatile network for parallel computation. Comput. Arch. Sst. 1981, 24, 300 309. 9. Xiang, Y.; Stewart, I.A. Augmented k-ar n-cubes. Inform. Sci. 2011, 181, 239 256. 10. Wu, J.; Huang, K. The balanced hpercube: A cube-based sstem for fault-tolerant applications. IEEE Trans. Comput. 1997, 46, 484 490. 11. Huang, K.; Wu, J. Fault-tolerant resource placement in balanced hpercubes. Inform. Sci. 1997, 99, 159 172. 12. Xu, M.; Hu, H.; Xu, J. Edge-pancclicit and Hamiltonian laceabilit of the balanced hpercubes. Appl. Math. Comput. 2007, 189, 1393 1401. 13. Yang, M. Bipanconnectivit of balanced hpercubes. Comput. Math. Appl. 2010, 60, 1859 1867. 14. Huang, K.; Wu, J. Area efficient laout of balanced hpercubes. Int. J. High Speed Electr. Sst. 1995, 6, 631 645. 15. Yang, M. Super connectivit of balanced hpercubes. Appl. Math. Comput. 2012, 219, 970 975. 16. Lü, H.; Li, X.; Zhang, H. Matching preclusion for balanced hpercubes. Theor. Comput. Sci. 2012, 465, 10 20, doi:10.1016/j.tcs.2012.09.020. 17. Lü, H.; Zhang, H. Hper-Hamiltonian laceabilit of balanced hpercubes. J. Supercomput. 2014, 68, 302 314, doi:10.1007/s11227-013-1040-6. 18. Lü, H.; Gao, X.; Yang, X. Matching etendabilit of balanced hpercubes. Ars Combinatoria 2016, 129, 261 274. 19. Lü, H. On etra connectivit and etra edge-connectivit of balanced hpercubes. Int. J. Comput. Math. 2017, 94, 813 820. 20. Zhou, J.-X.; Wu, Z.-L.; Yang, S.-C.; Yuan, K.-W. Smmetric propert and reliabilit of balanced hpercube. IEEE Trans. Comput. 2015, 64, 876 881. 21. Zhou, J.-X.; Kwak, J.; Feng, Y.-Q.; Wu, Z.-L. Automorphism group of the balanced hpercube. Ars Math. Contemp. 2017, 12, 145 154. 22. Jha, P.K.; Prasad, R. Hamiltonian decomposition of the rectangular twisted torus. IEEE Trans. Comput. 2012, 23, 1504 1507 23. Chang, N.-W.; Tsai, C.-Y.; Hsieh, S.-Y. On 3-etra connectivit and 3-etra edge connectivit of folded hpercubes. IEEE Trans. Comput. 2014, 63, 1594 1600. 24. Hsieh, S.-Y.; Yu, P.-Y. Fault-free mutuall independent Hamiltonian ccles in hpercubes with fault edges. J. Combin. Optim. 2007, 13, 153 162.
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