Hamiltonian paths and cycles pass through prescribed edges in the balanced hypercubes

Discrete Applied Mathematics 262 (2019) 56–71

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Hamiltonian paths and cycles pass through prescribed edges in the balanced hypercubes Dongqin Cheng Department of Mathematics, Jinan University, 510632, Guangzhou, China

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Article history: Received 16 July 2017 Received in revised form 13 December 2018 Accepted 22 February 2019 Available online 14 March 2019 Keywords: Interconnection network Balanced hypercube Hamiltonian path Hamiltonian cycle Prescribed edges

a b s t r a c t The n-dimensional balanced hypercube BHn (n ≥ 1) has been proved to be a bipartite graph. Let P be a set of edges in BHn . For any two vertices u, v from different partite sets of V (BHn ). In this paper, we prove that if |P | ≤ 2n − 2 and the subgraph induced by P has neither u nor v as internal vertices , or both of u and v as end-vertices, then BHn contains a Hamiltonian path joining u and v passing through every edge of P if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths. As a corollary, if |P | ≤ 2n − 1, then BHn contains a Hamiltonian cycle passing through every edge of P if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths. © 2019 Elsevier B.V. All rights reserved.

1. Introduction In parallel and distributed computer systems, the processors are connected according to a given interconnection network. An interconnection network is usually represented by a simple graph G = (V , E), where the vertex set V represents the set of processors and the edge set E represents the set of links between processors. Interconnection networks are widely studied by researchers [3,7–11,13,14,20,25]. The balanced hypercubes, proposed by Huang and Wu [12], are variants of the well-known hypercubes and possess superior properties over the hypercubes such as (i) the balanced hypercubes support an efficient reconfiguration without changing the adjacency relationship among tasks, and (ii) an odd-dimensional balanced hypercube has a smaller diameter than the comparable hypercube [22]. Many new properties of balanced hypercubes have been investigated recently, such as path and cycle embedding properties [4–6], edge-bipancyclicity, Hamiltonian laceability and Hyper-Hamiltonian laceability [15,17,23], matching preclusion [16], and symmetric property and reliability [28]. A Hamiltonian path (respectively, Hamiltonian cycle) in graph G is a path (respectively, cycle) that contains every vertex of V (G) exactly once. A graph is Hamiltonian if it contains a Hamiltonian cycle. The Hamiltonian property is a major requirement in designing network topologies since a topology structure containing Hamiltonian paths or cycles can efficiently simulate many algorithms designed on linear arrays or rings [1]. Each link in a parallel distributed system may be assigned distinct bandwidth, so it is meaningful to study the problem of how to embed a Hamiltonian cycle passing through prescribed edges into a network [2]. For a given set of edges P , a Hamiltonian path P (respectively, Hamiltonian cycle C ) passes through P if P (respectively, C ) contains every edge of P . Wang and Chen [18] proved that for a given edge set |E0 | and a faulty edge set F with 1 ≤ |E0 | ≤ 2n − 3 and |F | < n − (⌊|E0 |/2⌋ + 1), if the subgraph induced by E0 is pairwise vertex-disjoint paths, then the n-dimensional hypercube Qn (n ≥ 2) contains a fault-free Hamiltonian cycle passing through every edge of E0 . Chen et al. [2] obtained a lower bound for the number of Hamiltonian cycles E-mail address: [email protected]. https://doi.org/10.1016/j.dam.2019.02.033 0166-218X/© 2019 Elsevier B.V. All rights reserved.

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passing through a given edge in an n-dimensional crossed cube. Zhang and Zhang [26] proved that for a given edge set P and a faulty edge set F with |P | ≤ 2n − 2 and |F | ≤ 2n − (|P | + 2), a k-ary n-cube Qnk with n ≥ 2 and k ≥ 3 contains a Hamiltonian cycle passing through every edge of P if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths. Wang et al. [19] proved that for a given edge set P with at most 2n − 1 (n ≥ 2) edges, the 3-ary n-cube contains a Hamiltonian cycle passing through every edge of P if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths. Wang et al. [21] proved that for an edge set P with at most 2n − 1 edges, the k-ary n-cube contains a Hamiltonian cycle passing through every edge of P if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths. Inspired by the above results, in this paper, we consider Hamiltonian cycle and Hamiltonian path passing through a given set of edges in the n-dimensional balanced hypercube BHn . We prove that for a given edge set P with |P | ≤ 2n − 2, for any two vertices u and v from different partite sets of V (BHn ), BHn contains a Hamiltonian path between u and v passing through every edge of P if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths. Furthermore, if |P | ≤ 2n − 1, then BHn contains a Hamiltonian cycle passing through every edge of P if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths. The rest of this paper is organized as follows. In Section 2, we introduce some terminologies, definitions of BHn and some properties in BHn . In Section 3, we prove our main results. At last, we make conclusions in Section 4. 2. Preliminary A simple graph is denoted by G = (V , E), where V is vertex set and E is edge set. If (u, v ) ∈ E, then u and v are adjacent and they are each other’s neighbor. A path P joining u and v is denoted by P [u, v]. More precisely, P [u, v] = ⟨u(= u0 ), u1 , u2 , . . . , un−1 , v (= un )⟩, where the vertices are distinct and every two consecutive vertices are adjacent. u and v are called end-vertices of P [u, v]. The length of path P [u, v] is denoted by ℓ(P [u, v]). If the two end-vertices are the same and the length is at least three, then a path is called a cycle. ℓ-cycle is a cycle of length ℓ. A bipartite graph G = (V0 ∪ V1 , E) is a graph such that V0 ∩ V1 = ∅, and each edge with two end-vertices from V0 and V1 , respectively. A bipartite graph is Hamiltonian laceable if for any two vertices from different partite sets, there is a Hamiltonian path joining them. A bipartite graph is edge-bipancyclic if for any edge e, there are cycles of every even length ℓ with 4 ≤ ℓ ≤ |V (G)| containing e. Two graphs G and H are called isomorphic if there is a one-to-one and onto function π : V (G) → V (H) such that (u, v ) ∈ E(G) if and only if (π (u), π (v )) ∈ E(H). Moreover, if G = H, then G and H are called automorphic. A graph G is edge-transitive (respectively, vertex-transitive) if there is an automorphism between any two edges (respectively, vertices). Let u, v be two vertices in BHn and ⟨P ⟩ be a graph induced by P . {u, v} and P are called compatible in BHn if both u and v are incident to at most one edge of P and there is no path in ⟨P ⟩ joining u and v [19]. In a bipartite graph, for any two vertices u and v , and a set of edges P with |P | ≤ k, if {u, v} and P are compatible, and there is a Hamiltonian path joining u and v passing through every edge of P , then we say G is k-prescribed Hamiltonian laceable. Obviously, a 0-prescribed Hamiltonian laceable graph is Hamiltonian laceable. There are following two definitions for an n-dimensional balanced hypercube BHn . Definition 1 ([22]). An n-dimensional balanced hypercube BHn has vertex set {(a0 , a1 , . . . , an−1 ) | ai ∈ {0, 1, 2, 3} for i ∈ {0, 1, 2, . . . , n − 1}}. Each vertex (a0 , a1 , . . . , an−1 ) is adjacent to the following 2n vertices, ((a0 ± 1) mod 4, a1 , . . . , ai−1 , ai , ai+1 , . . . , an−1 ), and ((a0 ± 1) mod 4, a1 , . . . , ai−1 , (ai + (−1)a0 ) mod 4, ai+1 , . . . , an−1 ), where 1 ≤ i ≤ n − 1. In vertex u = (u0 , u1 , u2 , . . . , un−1 ) ∈ V (BHn ), u0 is called inner index, and ui s (1 ≤ i ≤ n − 1) are called i-dimensional index [27]. For an edge e = (u, v ) ∈ E(BHn ), if u and v differ the inner index, then e is called along dimension 0 and e is 0-dimensional edge, if u and v differ both the inner index and the i-dimensional index, then e is called along dimension i and e is called i-dimensional edge [27]. Definition 2 ([22]). An n-dimensional balanced hypercube BHn can be recursively defined. (1) BH1 is a 4-cycle denoted by ⟨(0), (1), (2), (3), (0)⟩. (2) For n ≥ 2, BHn is constructed by BHn0−1 , BHn1−1 , BHn2−1 , and BHn3−1 . Each vertex (a0 , a1 , . . . , an−2 , i) in BHni −1 (i ∈ {0, 1, 2, 3}) has two extra neighbors: (a) (a0 ± 1 (mod 4), a1 , . . ., an−2 , i + 1 (mod 4)) in BHni+−11 if a0 is even, (b) (a0 ± 1 (mod 4), a1 , . . ., an−2 , i − 1 (mod 4)) in BHni−−11 if a0 is odd. The illustrations of BH1 and BH2 can be found in Fig. 1. Throughout this paper, we use xi to denote xi ∈ V (BHni −1 ), where i ∈ {0, 1, 2, 3}. Let ∂ Dd be the set of edges along dimension d, 0 ≤ d ≤ n − 1 [27]. Lemma 1 ([27]). Let n ≥ 2 be an integer. Then BHn − ∂ Dd has four components, and each component is isomorphic to BHn−1 , where 0 ≤ d ≤ n − 1. d,i

In this paper, for a specified d, we use BHn−1 (i = 0, 1, 2, 3) to denote those four sub-balanced hypercubes. We use

i,j E(BHn−1 ) 2,3 E(BHn−1 ) is BHni −1 .

d,i

d,j

0,1

1 ,2

to denote the edges between sub-balanced hypercubes BHn−1 and BHn−1 . Let Ec = E(BHn−1 ) ∪ E(BHn−1 ) ∪ ∪ E(BHn3−,01 ), i.e., Ec is the set of edges between adjacent sub-balanced hypercubes. If d = n − 1, then BHnd−,i 1

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Fig. 1. The illustrations of BH1 and BH2 .

Lemma 2 ([22]). The balanced hypercube is bipartite. By Lemma 2, the vertices in BHn can be colored in two colors. Let the vertices in {u = (a0 , a1 , a2 , . . . , un−1 ) ∈ V (BHn )|a0 is even} be colored by white, and the vertices in {u = (a0 , a1 , a2 , . . . , un−1 ) ∈ V (BHn )|a0 is odd} be colored by black. Lemma 3 ([28]). The balanced hypercube is edge-transitive. Lemma 4 ([23]). The balanced hypercube BHn is Hamiltonian laceable for n ≥ 1. Lemma 5 ([23]). The balanced hypercube BHn is edge-bipancyclic. Lemma 6 ([24]). There are 22n−2 edges between BHni −1 and BHni+−11 for each 0 ≤ i ≤ 3, where n ≥ 2. Lemma 7 ([5]). Let X and Y be two partite sets of BHn , and x, u ∈ X , y, v ∈ Y . Then there exist two vertex-disjoint paths P [x, y] and R[u, v] in BHn , and V (P [x, y]) ∪ V (R[u, v]) = V (BHn ), where n ≥ 1. Lemma 8 ([4]). Let e be an edge in an n-dimensional balanced hypercube BHn , where n ≥ 2. Then there exist 2n − 2 cycles of length 8 containing e in BHn such that these cycles are edge-disjoint but e and each cycle is with only one edge in every sub-balanced hypercube BHni −1 for i = 0, 1, 2, 3. 3. Main results Throughout this paper, let P ⊂ E(BHn ) be a set of edges and u, v be two vertices of BHn from different partite sets of V (BHn ) such that P ∪{u, v} are compatible. Assume that BHn is partitioned along some dimension d into four sub-balanced d,i hypercubes. Let Pi = P ∩ E(BHn−1 ) where i = 0, 1, 2, 3 and Pc = P ∩ Ec . Lemma 9.

If |P | ≤ 2n − 2, then BHn can be divided along some dimension d into four BHn−1 s, and |Pc | ≤ 1.

Proof. The proof of this lemma is by contradiction. Since there are n dimensions in BHn . By Lemma 1, BHn can be partitioned along some dimension d into four sub-balanced hypercubes and each sub-balanced hypercube is isomorphic to BHn−1 . For any choice of d in {0, 1, 2, . . . , n − 1}, if Pc still contains at least two edges of P , then there are at least 2n edges in P . It is a contradiction to the assumption of P . By Lemma 9, BHn can be divided into four sub-balanced hypercubes. For simplicity, we may assume that d = n − 1. Then these four sub-balanced hypercubes are BHn0−1 , BHn1−1 , BHn2−1 , BHn3−1 and |Pc | ≥ 1. Without loss of generality, we may assume that |P0 | = max{|Pi | | i = 0, 1, 2, 3}. The following lemmas are based on this partition. We always assume that u is a white vertex and v is a black vertex. Lemma 10.

If |P0 | = 2n − 3, then BHn0−1 contains a Hamiltonian cycle passing through P0 .

Proof. The proof is by inductive hypothesis. Assume that BHni −1 is (2n − 4)-prescribed Hamiltonian laceable for i = 0, 1, 2, 3 and n ≥ 2. Since BH1i ∼ = BH1 , by Lemma 4 BH1i is 0-prescribed Hamiltonian laceable for n = 2. Assume that (a0 , b0 ) ∈ P0 , where a0 and b0 are incident to at most two edges of P0 . So {a0 , b0 } and P0 \ {(a0 , b0 )} are compatible in BHn0−1 . By assumption, BHn0−1 is (2n − 4)-prescribed Hamiltonian laceable, BHn0−1 contains a Hamiltonian path P [a0 , b0 ] passing through P0 \ {(a0 , b0 )}. Hence, P [a0 , b0 ] + (a0 , b0 ) is a Hamiltonian cycle passing through P0 in BHn0−1 .

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Fig. 2. The illustration of Lemma 11.

Fig. 3. The illustrations Cases 1, 2.1 and 2.2.2 in Lemma 12.

Lemma 11. n ≥ 2.

If |P0 | = 2n − 2, then BHn0−1 contains a Hamiltonian cycle or a Hamiltonian path passing through P0 , where

Proof. The proof is by inductive hypothesis. Assume that BHni −1 is (2n − 4)-prescribed Hamiltonian laceable for i = 0, 1, 2, 3 and n ≥ 2. Since BH1i ∼ = BH1 , by Lemma 4 BH1i is 0-prescribed Hamiltonian laceable for n = 2. Assume that (a0 , b0 ) ∈ P0 and b0 is an end-vertex of a path in the subgraph induced by P0 . By Lemma 10, BHn0−1 contains a Hamiltonian cycle C0 passing through P0 \ {(a0 , b0 )}. If (a0 , b0 ) ∈ C0 , then C0 is the desired cycle. If (a0 , b0 ) ̸ ∈ C0 , assume that c0 and d0 are neighbors of a0 and b0 respectively, such that two paths in C0 between a0 and b0 contain only one vertex of {c0 , d0 }. Assume that (a0 , c0 ), (b0 , d0 ) ∈ E(C0 ). Hence, C0 − (a0 , c0 ) − (b0 , d0 ) + (a0 , b0 ) is a Hamiltonian path passing through P0 in BHn0−1 . See Fig. 2. Lemma 12. If |Pc | = 0, |P | ≤ 2n − 2, and u, v are in the same sub-balanced hypercube, then BHn contains a Hamiltonian path P [u, v] passing through P , where n ≥ 2. Proof. The proof is by inductive hypothesis. Assume that BHni −1 is (2n − 4)-prescribed Hamiltonian laceable for i = 0, 1, 2, 3 and n ≥ 2. Since BH1i ∼ = BH1 , by Lemma 4 BH1i is 0-prescribed Hamiltonian laceable for n = 2. Case 1. |P0 | ≤ 2n − 4. In this case, |Pi | ≤ |P0 | ≤ 2n − 4 for i = 1, 2, 3. By symmetry of BHn , we only need to consider u, v ∈ V (BHn0−1 ). The cases of u, v ∈ V (BHni −1 ) for i = 1, 2, 3 are with similar discussions. Since |P0 | ≤ 2n − 4, by assumption, BHn0−1 is (2n − 4)-prescribed Hamiltonian laceable, there is a Hamiltonian path P0 [u, v] in BHn0−1 passing through P0 . Since ℓ(P0 [u, v]) − |P | = 22(n−1) − (2n − 2) ≥ 2 for n ≥ 2, we can 0 ,1 0,3 find an edge, denoted by (s0 , t0 ) ∈ E(P0 [u, v]), such that (s0 , t0 ) ̸ ∈ P0 and (s0 , s1 ) ∈ E(BHn−1 ), (t0 , t3 ) ∈ E(BHn−1 ). 1,2 2,3 By Lemma 6, there are two edges, denoted by (t1 , t2 ) ∈ E(BHn−1 ) and (s2 , s3 ) ∈ E(BHn−1 ). Since |Pi | ≤ 2n − 4 for i = 1, 2, 3, by assumption, BHni −1 is (2n − 4)-prescribed Hamiltonian laceable, there are Hamiltonian paths P [s1 , t1 ] passing through P1 in BHn1−1 , P [s2 , t2 ] passing through P2 in BHn2−1 , and P [s3 , t3 ] passing through P3 in BHn3−1 . Hence, P [u, v] = P0 [u, v] − (s0 , t0 ) + (s0 , s1 ) + P [s1 , t1 ] + (t1 , t2 ) + P [t2 , s2 ] + (s2 , s3 ) + P [s3 , t3 ] + (t3 , t0 ) is a Hamiltonian path passing through P in BHn . See Fig. 3(a).

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Case 2. |P0 | = 2n − 3. In this case, there is at most one edge of P , denoted by e′ , in E(BHn1−1 ) ∪ E(BHn2−1 ) ∪ E(BHn3−1 ). Without loss of generality, we assume that e′ ∈ E(BHn1−1 ). Then |P2 | = |P3 | = 0. (The cases of e′ ∈ E(BHn2−1 ) and e′ ∈ E(BHn3−1 ) can be proved similarly.) Case 2.1. u, v ∈ V (BHn0−1 ). By Lemma 10, there is a Hamiltonian cycle C0 passing through P0 . Clearly, u, v ∈ V (C0 ). Since {u, v}∪ P are compatible. u and v have neighbors in C0 , denoted by s0 and t0 , respectively, such that the two paths between u and v contain only one vertex of {s0 , t0 } respectively. By Definition 2, there are edges (t0 , t1 ), (s0 , s3 ) ∈ Ec . By Lemma 6, there is an 2,3 edge (t2 , t3 ) ∈ E(BHn−1 ). By assumption, e′ ∈ E(BHn1−1 ). By Lemma 5, e′ is contained in a Hamiltonian cycle C1 . Clearly, 1,2 t1 ∈ V (C1 ). Assume that (t1 , s1 ) ∈ C1 and (t1 , s1 ) ̸ = e′ . By Definition 2, there is an edge (s1 , s2 ) ∈ E(BHn−1 ). Since 3 2 |P2 | = |P3 | = 0, by Lemma 4, there are Hamiltonian paths P [s2 , t2 ] in BHn−1 and P [s3 , t3 ] in BHn−1 . Let P [u, v] = C0 − (u, s0 ) − (v, t0 ) + (t0 , t1 ) + C1 − (t1 , s1 ) + (s1 , s2 ) + P [s2 , t2 ] + (t2 , t3 ) + P [t3 , s3 ] + (s3 , s0 ). Then P [u, v] is a Hamiltonian path passing through P in BHn . See Fig. 3(b). Case 2.2. u, v ̸ ∈ V (BHn0−1 ). Since |P0 | = 2n − 3, by Lemma 10, BHn0−1 contains a Hamiltonian cycle C0 passing through P0 . Since ℓ(C0 ) − |P | ≥ 2(n−1) 2 − (2n − 2) ≥ 2 for n ≥ 2 and by Definition 2, each vertex of BHn0−1 has two neighbors in BHn1−1 , we can find an 0,3 0,1 edge, denoted by (s0 , t0 ) ∈ E(C0 ), such that (s0 , t0 ) ̸ ∈ P0 , (s0 , s1 ) ∈ E(BHn−1 ), (t0 , t3 ) ∈ E(BHn−1 ), s1 ̸ = v , t3 ̸ = u, and s1 and t3 are not incident to edges of P . Case 2.2.1. u, v ∈ V (BHn1−1 ). By assumption, P1 = {e′ }. By Lemma 5, e′ is contained in a Hamiltonian cycle C1 in BHn1−1 . Clearly, s1 ∈ V (C1 ). Assume 1,2 that (s1 , t1 ) ∈ E(C1 ) such that (s1 , t1 ) ̸ = e′ . By Definition 2 and Lemma 6, there are two edges (t1 , t2 ) ∈ E(BHn−1 ), (s2 , s3 ) ∈ 2,3 2 E(BHn−1 ). Since |P2 | = |P3 | = 0, by Lemma 4, there are Hamiltonian paths P [t2 , s2 ] in BHn−1 and P [t3 , s3 ] in BHn3−1 . If (u, v ) ∈ E(C1 ), let P [u, v] = C1 − (s1 , t1 ) − (u, v ) + C0 − (s0 , t0 ) + (t0 , t3 ) + P [t3 , s3 ] + (s3 , s2 ) + P [s2 , t2 ] + (t2 , t1 ). Hence, P [u, v] is a Hamiltonian path passing through P in BHn . If (u, v ) ̸ ∈ E(C1 ), assume that a1 and b1 are neighbors of u and v respectively in C1 . By Definition 2, there are edges (a1 , a0 ) ∈ E(BHn0−,11 ) and (b1 , b2 ) ∈ E(BHn1−,21 ). Clearly, a0 ∈ V (C0 ). 0,3 Assume that (a0 , c0 ) ∈ E(C0 ) such that (a0 , c0 ) ̸ ∈ P0 . By Definition 2 and Lemma 6, there are edges (c0 , c3 ) ∈ E(BHn−1 ), 1,2 2,3 2 ,3 (t1 , t2 ) ∈ E(BHn−1 ), (a2 , a3 ) ∈ E(BHn−1 ), and (d2 , d3 ) ∈ E(BHn−1 ). Since |P2 | = |P3 | = 0, by Lemma 7, there are vertexdisjoint paths P [t2 , d2 ] and P [b2 , a2 ] in BHn2−1 such that V (P [t2 , d2 ]) ∪ V (P [a2 , b2 ]) = V (BHn2−1 ), and P [a3 , t3 ] and P [c3 , d3 ] in BHn3−1 such that V (P [a3 , t3 ]) ∪ V (P [c3 , d3 ]) = V (BHn3−1 ). Denote C0 = P [a0 , s0 ] + (s0 , t0 ) + P [t0 , c0 ] + (c0 , a0 ) and C1 = P [u, s1 ] + (s1 , t1 ) + P [t1 , v] + (v, b1 ) + P [b1 , a1 ] + (a1 , u). Let P [u, v] = P [u, s1 ] + (s1 , s0 ) + P [s0 , a0 ] + (a0 , a1 ) + P [a1 , b1 ]+(b1 , b2 )+P [b2 , a2 ]+(a2 , a3 )+P [a3 , t3 ]+(t3 , t0 )+P [t0 , c0 ]+(c0 , c3 )+P [c3 , d3 ]+(d3 , d2 )+P [d2 , t2 ]+(t2 , t1 )+P [t1 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . Case 2.2.2. u, v ∈ V (BHn2−1 ). By assumption, |P2 | = 0, by Lemma 4, BHn2−1 contains a Hamiltonian path P2 joining u and v . With the same discussion as Case 2.2.1, we can find a Hamiltonian cycle C1 in BHn1−1 and an edge (s1 , t1 ) ∈ E(C1 ) such that (s1 , t1 ) ̸ = e′ , 1,2 2 ,3 and an edge (t1 , t2 ) ∈ E(BHn−1 ). Clearly, t2 ∈ V (P2 ). Assume that (t2 , s2 ) ∈ E(P2 ) and (s2 , s3 ) ∈ E(BHn−1 ). Denote 3 P [u, v] = P [u, t2 ] + (t2 , s2 ) + P [s2 , v]. Since |P3 | = 0, by Lemma 4, BHn−1 contains a Hamiltonian path P [t3 , s3 ]. Let P [u, v] = P [u, t2 ] + (t2 , t1 ) + C1 − (s1 , t1 ) + (s1 , s0 ) + C0 − (s0 , t0 ) + (t0 , t3 ) + P [t3 , s3 ] + (s3 , s2 ) + P [s2 , v]. Then P [u, v] is a Hamiltonian path passing through P in BHn . See Fig. 3(c). Case 2.2.3. u, v ∈ V (BHn3−1 ). Since |P3 | = 0, by Lemma 4, BHn3−1 contains a Hamiltonian path P3 [u, v]. Clearly, t3 ∈ P3 . Assume that (t3 , s3 ) ∈ P3 and 2,3 (s3 , s2 ) ∈ E(BHn−1 ). Denote P3 [u, v] = P [u, t3 ] + (t3 , s3 ) + P [s3 , v]. With the same discussion as Case 2.2.1, we can find 1 ,2 a Hamiltonian cycle C1 in BHn1−1 , and an edge (s1 , t1 ) ∈ E(C1 ) such that (s1 , t1 ) ̸ = e′ , and edge (t1 , t2 ) ∈ E(BHn−1 ). Since 2 |P2 | = 0, by Lemma 4, BHn−1 contains a Hamiltonian path P [t2 , s2 ]. Let P [u, v] = P [u, t3 ] + (t3 , t0 ) + C0 − (t0 , s0 ) + (s0 , s1 ) + C1 − (s1 , t1 ) + (t1 , t2 ) + P [t2 , s2 ] + (s2 , s3 ) + P [s3 , v]. Then P [u, v] is a Hamiltonian path passing through P in BHn . Case 3. |P0 | = 2n − 2. In this case, P = P0 and there are no edges of P outside of BHn0−1 . By Lemma 11, BHn0−1 contains a Hamiltonian path P [a0 , b0 ] passing through P0 . (For easy to discuss, if BHn0−1 contains a Hamiltonian cycle, then we let P [a0 , b0 ] ∪ (a0 , b0 ) 0 ,1 0,3 be the desired Hamiltonian cycle.) By Definition 2, there are edges (a0 , a1 ) ∈ E(BHn−1 ) and (b0 , b3 ) ∈ E(BHn−1 ). 0 Case 3.1. u, v ∈ V (BHn−1 ). Clearly, u, v ∈ V (P [a0 , b0 ]). (Without loss of generality, we assume that (u, v ) ̸ ∈ E(P [a0 , b0 ]), otherwise it is more easier and can be proved with the similar idea.) Assume that s0 and t0 are neighbors of u and v on P [a0 , b0 ], respectively, such that (u, s0 ), (v, t0 ) ̸ ∈ P0 and the paths between u and v in P [a0 , b0 ] contain only one vertex of {s0 , t0 }. Denote P [a0 , b0 ] = P [a0 , u] + (u, s0 ) + P [s0 , v] + (v, t0 ) + P [t0 , b0 ]. By Lemma 6, we can find edges (b1 , b2 ) ∈ 1,2 1,2 2,3 2,3 E(BHn−1 ), (s1 , s2 ) ∈ E(BHn−1 ), (t2 , t3 ) ∈ E(BHn−1 ), (a2 , a3 ) ∈ E(BHn−1 ). By Lemma 7, there exist two node-disjoint paths 1 P [a1 , b1 ] and P [t1 , s1 ] in BHn−1 such that V (P [a1 , b1 ]) ∪ V (P [s1 , t1 ]) = V (BHn1−1 ), P [a2 , b2 ] and P [t2 , s2 ] in BHn2−1 such that V (P [a2 , b2 ]) ∪ V (P [t2 , s2 ]) = V (BHn2−1 ), and P [a3 , b3 ] and P [s3 , t3 ] in BHn3−1 such that V (P [a3 , b3 ]) ∪ V (P [s3 , t3 ]) = V (BHn3−1 ). Let P [u, v] = P [u, a0 ] + (a0 , a1 ) + P [a1 , b1 ] + (b1 , b2 ) + P [b2 , a2 ] + (a2 , a3 ) + P [a3 , b3 ] + (b3 , b0 ) + P [b0 , t0 ] + (t0 , t1 ) +

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Fig. 4. The illustrations of Cases 3.1, 3.2, and 3.3 in Lemma 12.

P [t1 , s1 ] + (s1 , s2 ) + P [s2 , t2 ] + (t2 , t3 ) + P [t3 , s3 ] + (s3 , s0 ) + P [s0 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . See Fig. 4(a). Case 3.2. u, v ∈ V (BHn1−1 ). 1,2 2,3 Since |P2 | = |P3 | = 0, by Lemma 6, we find two edges (b1 , b2 ) ∈ E(BHn−1 ) and (a2 , a3 ) ∈ E(BHn−1 ). Since 3 2 |P2 | = |P3 | = 0, by Lemma 4, there are Hamiltonian paths P [a2 , b2 ] in BHn−1 and P [a3 , b3 ] in BHn−1 . By Lemma 7, there are vertex-disjoint paths P [u, a1 ] and P [v, b1 ] in BHn1−1 such that V (P [u, a1 ]) ∪ V (P [v, b1 ]) = V (BHn1−1 ). Let P [u, v] = P [u, a1 ] + (a1 , a0 ) + P [a0 , b0 ] + (b0 , b3 ) + P [b3 , a3 ] + (a3 , a2 ) + P [a2 , b2 ] + (b2 , b1 ) + P [b1 , v]. Then P [u, v] is a Hamiltonian path passing through P in BHn . See Fig. 4(b). Case 3.3. u, v ∈ V (BHn2−1 ). 1,2 2,3 Since |P2 | = 0, by Lemma 6 we can find two edges (b1 , b2 ) ∈ E(BHn−1 ) and (a2 , a3 ) ∈ E(BHn−1 ). Since |P1 | = |P3 | = 0, 3 1 by Lemma 4, there are Hamiltonian paths P [a1 , b1 ] in BHn−1 and P [a3 , b3 ] in BHn−1 . Since |P2 | = 0, by Lemma 7, there are two vertex-disjoint paths P [u, b2 ] and P [v, a2 ] in BHn2−1 such that V (P [u, b2 ]) ∪ V (P [v, a2 ]) = V (BHn2−1 ). Let P [u, v] = P [u, b2 ] + (b2 , b1 ) + P [b1 , a1 ] + (a1 , a0 ) + P [a0 , b0 ] + (b0 , b3 ) + P [b3 , a3 ] + (a3 , a2 ) + P [a2 , v]. Then P [u, v] is a Hamiltonian path passing through P in BHn . See Fig. 4(c). Case 3.4. u, v ∈ V (BHn3−1 ). 2,3 1 ,2 By Lemma 6, we can find two edges (b1 , b2 ) ∈ E(BHn−1 ) and (a2 , a3 ) ∈ E(BHn−1 ). Since |P1 | = |P2 | = 0, by 1 2 Lemma 4, there are Hamiltonian paths P [a1 , b1 ] in BHn−1 and P [a2 , b2 ] in BHn−1 . Since |P3 | = 0, by Lemma 7, there are vertex-disjoint paths P [u, a3 ] and P [v, b3 ] in BHn3−1 such that V (P [u, a3 ]) ∪ V (P [v, b3 ]) = V (BHn3−1 ). Let P [u, v] = P [u, a3 ] + (a3 , a2 ) + P [a2 , b2 ] + (b2 , b1 ) + P [b1 , a1 ] + (a1 , a0 ) + P [a0 , b0 ] + (b0 , b3 ) + P [b3 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . Lemma 13. If |Pc | = 0, |P | ≤ 2n − 2, and u, v are in different sub-balanced hypercubes, then BHn contains a Hamiltonian path P [u, v] passing through P , where n ≥ 2. Proof. The proof is by inductive hypothesis. Assume that BHni −1 is (2n − 4)-prescribed Hamiltonian laceable for i = 0, 1, 2, 3 and n ≥ 2. Since BH1i ∼ = BH1 , by Lemma 4 BH1i is 0-prescribed Hamiltonian laceable for n = 2. Case 1. |P0 | ≤ 2n − 4. In this case, |Pi | ≤ 2n − 4 for i = 0, 1, 2, 3, 4. Case 1.1. u and v are in adjacent sub-balanced hypercubes. By symmetry of BHn , we only consider u ∈ V (BHn0−1 ) and v ∈ V (BHn1−1 ). 0 ,3 2,3 1,2 By Lemma 6, we can find edges (s0 , s3 ) ∈ E(BHn−1 ), (t3 , t2 ) ∈ E(BHn−1 ), and (s1 , s2 ) ∈ E(BHn−1 ). Since |Pi | ≤ 2n − 4 i for i = 0, 1, 2, 3, by assumption, BHn−1 is (2n − 4)-prescribed Hamiltonian laceable, there are Hamiltonian paths P [u, s0 ] passing through P0 in BHn0−1 , P [s3 , t3 ] passing through P3 in BHn3−1 , P [s2 , t2 ] passing through P2 in BHn2−1 , and P [v, s1 ] passing through P1 in BHn1−1 . Let P [u, v] = P [u, s0 ] + (s0 , s3 ) + P [s3 , t3 ] + (t3 , t2 ) + P [t2 , s2 ] + (s2 , s1 ) + P [s1 , v]. Then P [u, v] is a Hamiltonian path passing through P in BHn . Case 1.2. u and v are in non-adjacent sub-balanced hypercubes. By symmetry of BHn , we only consider u ∈ V (BHn0−1 ) and v ∈ V (BHn2−1 ). Since |V (BHn3−1 )| = 22(n−1) and |P | = 2n − 2, there is a white vertex, say s3 ∈ V (BHn3−1 ), such that (s3 , s0 ), (s3 , w0 ) ∈ 0,3 E(BHn−1 ) and s0 , w0 are not incident to edges of P0 , s3 is not incident to edges of P3 . Furthermore, for each neighbor of s3 in 3 BHn−1 , its two neighbors in BHn2−1 are also not incident to edges of P2 . By assumption, BHn0−1 is (2n − 4)-prescribed Hamiltonian laceable, BHn0−1 contains a Hamiltonian path P [u, s0 ] passing through P0 . Clearly, w0 ∈ V (P [u, s0 ]). Assume that 0,1 2,3 (w0 , t0 ) ∈ E(P [u, s0 ]) and (t0 , t1 ) ∈ E(BHn−1 ). By Lemma 6, we can find an edge (t2 , t3 ) ∈ E(BHn−1 ). By assumption BHni −1 is

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Fig. 5. The illustrations of Cases 1.2, 2.2 and 2.3 in Lemma 13.

(2n − 4)-prescribed Hamiltonian laceable, BHn3−1 contains a Hamiltonian path P [s3 , t3 ] passing through P3 , and BHn2−1 contains a Hamiltonian path P [v, t2 ] passing through P2 . Assume that w3 is a neighbor of s3 in P [s3 , t3 ] and (w3 , w2 ) ∈ 1,2 2,3 E(BHn−1 ). Clearly, w2 ∈ V (P [v, t2 ]). Assume that (w2 , s2 ) ∈ E(P [v, t2 ]) and (s2 , s1 ) ∈ E(BHn−1 ). Since |P1 | ≤ 2n − 4, by 1 assumption, BHn−1 contains a Hamiltonian path P [s1 , t1 ] passing through P1 . Denote P [u, s0 ] = P [u, w0 ] + (w0 , t0 ) + P [t0 , s0 ], P [s3 , t3 ] = (s3 , w3 ) + P [w3 , t3 ] and P [v, t2 ] = P [v, w2 ] + (w2 , s2 ) + P [s2 , t2 ]. Let P [u, v] = P [u, w0 ] + (w0 , s3 ) + (s3 , s0 ) + P [s0 , t0 ] + (t0 , t1 ) + P [t1 , s1 ] + (s1 , s2 ) + P [s2 , t2 ] + (t2 , t3 ) + P [t3 , w3 ] + (w3 , w2 ) + P [w2 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . See Fig. 5(a). Case 2. |P0 | = 2n − 3. We only consider |P1 | = 1. Denote P1 = {e1 }. (The cases of |P2 | = 1 and |P3 | = 1 can be proved similarly.) By Lemma 10, BHn0−1 contains a Hamiltonian cycle C0 passing through P0 . Case 2.1. u ∈ V (BHn0−1 ), v ∈ V (BHn1−1 ). 0,3 Let (u, a0 ) ∈ E(C0 ) such that (u, a0 ) ̸ ∈ P0 . By Definition 2, there is an edge (a0 , a3 ) ∈ E(BHn−1 ). By Lemma 6, there is 2,3 an edge (b3 , b2 ) ∈ E(BHn−1 ). By Lemma 5, e1 is contained in a Hamiltonian cycle C1 . Assume that (v, a1 ) ∈ E(C1 ) such that 1,2 (v, a1 ) ̸ = e1 . By Definition 2, there is an edge (a1 , a2 ) ∈ E(BHn−1 ). Denote C0 = P [u, a0 ]+ (a0 , u) and C1 = P [v, a1 ]+ (a1 , v ). Since |P2 | = |P3 | = 0, by Lemma 4, BHn2−1 contains a Hamiltonian path P [a2 , b2 ] and BHn3−1 contains a Hamiltonian path P [a3 , b3 ]. Let P [u, v] = P [u, a0 ] + (a0 , a3 ) + P [a3 , b3 ] + (b3 , b2 ) + P [b2 , a2 ] + (a2 , a1 ) + P [a1 , v]. Then P [u, v] is a Hamiltonian path passing through P in BHn . Case 2.2. u ∈ V (BHn0−1 ), v ∈ V (BHn2−1 ). 0 ,1 By Lemma 5, e1 is contained in a Hamiltonian cycle C1 in BHn1−1 . Let d0 be a vertex in C0 such that (d0 , d1 ) ∈ E(BHn−1 ) 0,3 and d1 is not incident to e1 . Assume that (d0 , c0 ), (u, a0 ) ∈ E(C0 ), (d1 , c1 ) ∈ E(C1 ), and (c0 , c3 ), (a0 , a3 ) ∈ E(BHn−1 ), (c1 , c2 ) ∈ 2 ,3 1,2 E(BHn−1 ). By Lemma 6, we can find two edges (d3 , d2 ), (b3 , b2 ) ∈ E(BHn−1 ). Since |P3 | = |P2 | = 0, by Lemma 7, there are 3 two vertex-disjoint paths P [a3 , d3 ] and P [c3 , b3 ] in BHn−1 such that V (P [a3 , d3 ]) ∪ V (P [c3 , b3 ]) = V (BHn3−1 ), and P [c2 , d2 ] and P [b2 , v] in BHn2−1 such that V (P [d2 , c2 ]) ∪ V (P [b2 , v]) = V (BHn2−1 ). Denote C0 = P [u, d0 ] + (d0 , c0 ) + P [c0 , a0 ] + (a0 , u) and C1 = P [c1 , d1 ] + (d1 , c1 ). Clearly, e1 ∈ E(P [c1 , d1 ]). Let P [u, v] = P [u, d0 ] + (d0 , d1 ) + P [d1 , c1 ] + (c1 , c2 ) + P [c2 , d2 ] + (d2 , d3 ) + P [d3 , a3 ] + (a3 , a0 ) + P [a0 , c0 ] + (c0 , c3 ) + P [c3 , b3 ] + (b3 , b2 ) + P [b2 , v]. Then P [u, v] is a Hamiltonian path passing through P in BHn . See Fig. 5(b). Case 2.3. u ∈ V (BHn0−1 ), v ∈ V (BHn3−1 ). 0,3 Assume that (u, a0 ) ∈ E(C0 ) and (u, a0 ) ̸ ∈ P0 , furthermore, (a0 , a3 ) ∈ E(BHn−1 ). Since ℓ(C0 ) = 22(n−1) and 22(n−1) −|P0 |− |{(u, a0 )}| = 22(n−1) − (2n − 3) − 1 ≥ 2 for n ≥ 2, we can find an edge , say (c0 , b0 ), in E(C0 ) − P0 − (u, a0 ). By Definition 2, 0,3 0,1 2,3 there are edges (c0 , c3 ) ∈ E(BHn−1 ) and (b0 , b1 ) ∈ E(BHn−1 ). By Lemma 6, we can find an edge (b2 , b3 ) ∈ E(BHn−1 ). Denote C0 = P [u, b0 ] + (b0 , c0 ) + P [c0 , a0 ] + (a0 , u). Since |P1 | = 1 and P1 = {e1 }. By Lemma 5, e1 is contained in a Hamiltonian cycle C1 of BHn1−1 . Clearly, b1 ∈ V (C1 ). Assume that (b1 , a1 ) ∈ E(C1 ) such that (b1 , a1 ) ̸ = e1 , and 1,2 (a1 , a2 ) ∈ E(BHn−1 ). Since |P2 | = 0, by Lemma 4, BHn2−1 contains a Hamiltonian path P [a2 , b2 ]. Since |P3 | = 0, by 3 Lemma 7, BHn−1 contains two vertex-disjoint paths P [c3 , v] and P [a3 , b3 ] such that V (P [c3 , v]) ∪ V (P [a3 , b3 ]) = V (BHn3−1 ). Let P [u, v] = P [u, b0 ]+ (b0 , b1 ) + P [b1 , a1 ]+ (a1 , a2 ) + P [a2 , b2 ]+ (b2 , b3 ) + P [b3 , a3 ]+ (a3 , a0 ) + P [a0 , c0 ]+ (c0 , c3 ) + P [c3 , v]. Then P [u, v] is a Hamiltonian path passing through P in BHn . See Fig. 5(c). Case 2.4. u ∈ V (BHn1−1 ), v ∈ V (BHn2−1 ). By Lemma 10, BHn0−1 contains a Hamiltonian cycle C0 passing through P0 . Let Eu0 = {(a0 , b0 )|(u, a1 ) ∈ E(BHn1−1 ), (a0 , a1 ) ∈ 0,1 E(BHn−1 ), (a0 , b0 ) ∈ E(C0 )}. If n = 2, then by Lemma 5, e1 is contained in a Hamiltonian cycle C1 in BHn1−1 . Note that u has two neighbors in C1 , and by Definition 2, each neighbor has two neighbors in BHn0−1 . Since |P0 | = 1 for n = 2. u is incident to an edge, say (u, a1 ), not equal to e1 such that its corresponding edge in Eu0 is not in P0 . If n ≥ 3, since u is incident to 2n − 2 edges in BHn1−1 , there is at least one edge incident to u, denoted by (u, a1 ), such that

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Fig. 6. The illustrations of Cases 2.5, 3.1.1, and 3.3.1 in Lemma 13.

(u, a1 ) ̸ ∈ P1 and its corresponding edge in Eu0 is not in P0 . Since |P1 | = 1 and P1 = {e1 }, we denote e1 = (c1 , d1 ), where c1 is white vertex and d1 is black vertex. By Lemma 7, there are vertex-disjoint paths P [a1 , c1 ] and P [u, d1 ] in BHn1−1 such that VP [a1 , c1 ] ∪ V (P [u, d1 ]) = V (BHn1−1 ). Then C1 = (a1 , u) + P [u, d1 ] + (d1 , c1 ) + P [c1 , a1 ] is a Hamiltonian 0,3 cycle passing through P1 . By Definition 2 and Lemma 6, we can find two edges, denoted by (b0 , b3 ) ∈ E(BHn−1 ) and 2 ,3 3 2 (a3 , a2 ) ∈ E(BHn−1 ). By Lemma 4, BHn−1 contains a Hamiltonian path P [a3 , b3 ] and BHn−1 contains a Hamiltonian path P [v, a2 ]. Let P [u, v] = C1 − (u, a1 ) + (a1 , a0 ) + C0 − (a0 , b0 ) + (b0 , b3 ) + P [b3 , a3 ] + (a3 , a2 ) + P [a2 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . Case 2.5. u ∈ V (BHn1−1 ), v ∈ V (BHn3−1 ). If n = 2, then by Lemma 5, BHn1−1 contains a cycle C1 passing through P1 . Since |P0 | = 1 when n = 2, u is incident to 0,1 an edge, denoted by (u, a1 ), in C1 , such that (a1 , a0 ) ∈ E(BHn−1 ) and (a0 , b0 ) ∈ E(C0 ) and (a0 , b0 ) ̸ ∈ P0 . If n ≥ 3, since u is 0,1 1 incident to 2n − 2 edges in BHn−1 , u is incident to an edge, denoted by (u, a1 ), such that (a1 , a0 ) ∈ E(BHn−1 ) and a0 is not 1 incident to edges of P0 . Since |{(u, a1 )}|+|P1 | = 2 < (2n − 3) for n ≥ 3, by Lemma 10, BHn−1 contains a Hamiltonian cycle C1 passing through (u, a1 ) and P1 . In both cases, we can find an edge, denoted by (c1 , d1 ) ∈ E(C1 ) such that (c1 , d1 ) ̸ ∈ P1 , 0,1 and (d1 , d0 ) ∈ E(BHn−1 ) such that (d0 , c0 ) ∈ E(C0 ) and (c0 , d0 ) ̸ ∈ P0 . (Since ℓ(C1 ) − |{(u, a1 )}| − |P1 | = 22(n−1) − 2 > 2n − 3 ≥ |P0 | for n ≥ 2, we can find such edge (c1 , d1 ) in C1 .) Denote C1 = P [u, d1 ] + (d1 , c1 ) + P [c1 , a1 ] + (a1 , u) 0,3 and C0 = P [a0 , d0 ] + (d0 , c0 ) + P [c0 , b0 ] + (b0 , a0 ). By Definition 1, there are edges (c0 , c3 ) ∈ E(BHn−1 ), (b0 , b3 ) ∈ 0 ,3 1,2 2,3 E(BHn−1 ), (c1 , c2 ) ∈ E(BHn−1 ). By Lemma 6, there is an edge (d2 , d3 ) ∈ E(BHn−1 ). Since |P2 | = 0, by Lemma 4, BHn2−1 contains a Hamiltonian path P [c2 , d2 ]. Since |P3 | = 0, by Lemma 7, BHn3−1 contains two vertex-disjoint paths P [c3 , v] and P [b3 , d3 ] in BHn3−1 such that V (P [c3 , v]) ∪ V (P [b3 , d3 ]) = V (BHn3−1 ). Let P [u, v] = P [u, d1 ] + (d1 , d0 ) + P [d0 , a0 ] + (a0 , a1 ) + P [a1 , c1 ] + (c1 , c2 ) + P [c2 , d2 ] + (d2 , d3 ) + P [d3 , b3 ] + (b3 , b0 ) + P [b0 , c0 ] + (c0 , c3 ) + P [c3 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . See Fig. 6(a). Case 2.6. u ∈ V (BHn2−1 ), v ∈ V (BHn3−1 ). Since |P1 | = 1 and P1 = {e1 }, by Lemma 5, e1 is contained in a Hamiltonian cycle C1 in BHn1−1 . Since ℓ(C1 ) = 22(n−1) − |P | = 22(n−1) − (2n − 2) ≥ 2 for n ≥ 2, we can find an edge, denoted by (a1 , b1 ), in C1 such that (a1 , b1 ) ̸ ∈ P1 , and 0,1 1 ,2 (a1 , a0 ) ∈ E(BHn−1 ) such that (a0 , b0 ) ∈ E(C0 ) and (a0 , b0 ) ̸ ∈ P0 . By Definition 2, there are edges (b1 , b2 ) ∈ E(BHn−1 ) and 0,3 3 2 (b0 , b3 ) ∈ E(BHn−1 ). Since |P2 | = |P3 | = 0, by Lemma 4, BHn−1 contains a Hamiltonian path P [b2 , u] and BHn−1 contains a Hamiltonian path P [b3 , v]. Let P [u, v] = P [u, b2 ] + (b2 , b1 ) + C1 − (a1 , b1 ) + (a0 , a1 ) + C0 − (a0 , b0 ) + (b0 , b3 ) + P [b3 , v]. Hence, P [u, v] is a Hamiltonian path passing through P . Case 3. |P0 | = 2n − 2. In this case, |P1 | = |P2 | = |P3 | = 0. Since |P0 | = 2n − 2, by Lemma 11, BHn0−1 contains a Hamiltonian path P [a0 , b0 ] passing through P0 . (Note that, if BHn0−1 contains a Hamiltonian cycle, then (a0 , b0 ) is an edge.) By Lemma 6, there are edges 0,1 0,3 1,2 1,2 2,3 2 ,3 (a0 , a1 ) ∈ E(BHn−1 ), (b0 , b3 ) ∈ E(BHn−1 ), (b1 , b2 ) ∈ E(BHn−1 ), (c1 , c2 ) ∈ E(BHn−1 ), (a2 , a3 ) ∈ E(BHn−1 ), (d2 , d3 ) ∈ E(BHn−1 ). 0 Case 3.1. u ∈ V (BHn−1 ). 0,3 Assume that (u, c0 ) ∈ E(P [a0 , b0 ]), and (u, c0 ) ̸ ∈ P0 . By Definition 2, there is an edge (c0 , c3 ) ∈ E(BHn−1 ). Denote P [a0 , b0 ] = P [a0 , c0 ] + (c0 , u) + P [u, b0 ]. We consider the following three cases. Case 3.1.1. v ∈ V (BHn1−1 ). Since |P1 | = 0, by Lemma 7, BHn1−1 contains two vertex-disjoint paths P [a1 , b1 ] and P [v, c1 ] such that V (P [a1 , b1 ]) ∪ V (P [v, c1 ]) = V (BHn1−1 ), BHn2−1 contains two vertex-disjoint paths P [a2 , b2 ] and P [c2 , d2 ] such that V (P [a2 , b2 ]) ∪ V (P [c2 , d2 ]) = V (BHn2−1 ), and BHn3−1 contains two vertex-disjoint paths P [c3 , d3 ] and P [b3 , a3 ] such that V (P [c3 , d3 ]) ∪ V (P [b3 , a3 ]) = V (BHn3−1 ). Let P [u, v] = P [u, b0 ] + (b0 , b3 ) + P [b3 , a3 ] + (a3 , a2 ) + P [a2 , b2 ] + (b2 , b1 ) + P [b1 , a1 ] + (a1 , a0 ) + P [a0 , c0 ] + (c0 , c3 ) + P [c3 , d3 ] + (d3 , d2 ) + P [d2 , c2 ] + (c2 , c1 ) + P [c1 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . See Fig. 6(b).

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Case 3.1.2. v ∈ V (BHn2−1 ). By Lemma 7, BHn3−1 contains two vertex-disjoint paths P [c3 , d3 ] and P [b3 , a3 ] such that V (P [c3 , d3 ]) ∪ V (P [b3 , a3 ]) = V (BHn3−1 ), and BHn2−1 contains two vertex-disjoint paths P [a2 , c2 ] and P [d2 , v] such that V (P [v, d2 ]) ∪ V (P [a2 , c2 ]) = V (BHn2−1 ). Since |P1 | = 0, by Lemma 4, BHn1−1 contains a Hamiltonian path P [a1 , c1 ]. Let P [u, v] = P [u, b0 ] + (b0 , b3 ) + P [b3 , a3 ] + (a3 , a2 ) + P [a2 , c2 ] + (c2 , c1 ) + P [c1 , a1 ] + (a1 , a0 ) + P [a0 , c0 ] + (c0 , c3 ) + P [c3 , d3 ] + (d3 , d2 ) + P [d2 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . Case 3.1.3. v ∈ V (BHn3−1 ). By Lemma 7, BHn3−1 contains two vertex-disjoint paths P [c3 , v] and P [b3 , d3 ] such that V (P [c3 , v]) ∪ V (P [b3 , d3 ]) = V (BHn3−1 ). Since |P1 | = 0, by Lemma 4, BHn1−1 contains a Hamiltonian path P [a1 , c1 ] and BHn2−1 contains a Hamiltonian path P [c2 , d2 ]. Since |P3 | = 0, by Lemma 7, BHn3−1 contains two vertex-disjoint paths P [c3 , v] and P [b3 , d3 ] such that V (P [c3 , v]) ∪ V (P [b3 , d3 ]) = V (BHn3−1 ). Let P [u, v] = P [u, b0 ] + (b0 , b3 ) + P [b3 , d3 ] + (d2 , d2 ) + P [d2 , c2 ] + (c2 , c1 ) + P [c1 , a1 ] + (a1 , a0 ) + P [a0 , c0 ] + (c0 , c3 ) + P [c3 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . Case 3.2. u ∈ V (BHn1−1 ), v ∈ V (BHn2−1 ). Since |P1 | = |P2 | = |P3 | = 0, by Lemma 4, BHn1−1 contains a Hamiltonian path P [u, a1 ], BHn2−1 contains a Hamiltonian path P [d2 , v], and BHn3−1 contains a Hamiltonian path P [b3 , d3 ]. Let P [u, v] = P [u, a1 ] + (a1 , a0 ) + P [a0 , b0 ] + (b0 , b3 ) + P [b3 , d3 ] + (d3 , d2 ) + P [d2 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . Case 3.3. u ∈ V (BHn1−1 ), v ∈ V (BHn3−1 ) or u ∈ V (BHn2−1 ), v ∈ V (BHn3−1 ). Since ℓ(P [a0 , b0 ]) − |P0 | = 22(n−1) − 1 − (2n − 2) ≥ 1 for n ≥ 2, we can find at least one edge, denoted by (c0 , d0 ) ∈ E(P [a0 , b0 ]), such that (c0 , d0 ) ̸ ∈ P0 . Denote P [a0 , b0 ] = P [a0 , d0 ] + (d0 , c0 ) + P [c0 , b0 ]. By Definition 2, there are 0,3 0,1 edges (d0 , d1 ) ∈ E(BHn−1 ) and (c0 , c3 ) ∈ E(BHn−1 ). 3 1 Case 3.3.1. u ∈ V (BHn−1 ), v ∈ V (BHn−1 ). Since |P2 | = 0, by Lemma 4, BHn2−1 contains a Hamiltonian path P [c2 , d2 ]. Since |P1 | = 0, by Lemma 7, BHn1−1 contains two vertex-disjoint paths P [u, a1 ] and P [c1 , d1 ] such that V (P [u, a1 ]) ∪ V (P [c1 , d1 ]) = V (BHn1−1 ), and BHn3−1 contains two vertex-disjoint paths P [b3 , v] and P [c3 , d3 ] such that V (P [b3 , v]) ∪ V (P [c3 , d3 ]) = V (BHn3−1 ). Let P [u, v] = P [u, a1 ] + (a1 , a0 ) + P [a0 , d0 ] + (d0 , d1 ) + P [d1 , c1 ] + (c1 , c2 ) + P [c2 , d2 ] + (d2 , d3 ) + P [d3 , c3 ] + (c3 , c0 ) + P [c0 , b0 ] + (b0 , b3 ) + P [b3 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . See Fig. 6(c). Case 3.3.2. u ∈ V (BHn2−1 ), v ∈ V (BHn3−1 ). Since |P1 | = 0, by Lemma 7, BHn1−1 contains two vertex-disjoint paths P [a1 , c1 ] and P [d1 , b1 ] such that V (P [a1 , c1 ]) ∪ V (P [d1 , b1 ]) = V (BHn1−1 ), BHn2−1 contains two vertex-disjoint paths P [u, b2 ] and P [c2 , d2 ] such that V (P [u, b2 ]) ∪ V (P [c2 , d2 ]) = V (BHn2−1 ), and BHn3−1 contains two vertex-disjoint paths P [c3 , v] and P [b3 , d3 ] such that V (P [c3 , v]) ∪ V (P [b3 , d3 ]) = V (BHn3−1 ). Let P [u, v] = P [u, b2 ] + (b2 , b1 ) + P [b1 , d1 ] + (d1 , d0 ) + P [d0 , a0 ] + (a0 , a1 ) + P [a1 , c1 ] + (c1 , c2 ) + P [c2 , d2 ]+ (d2 , d3 ) + P [d3 , b3 ]+ (b3 , b0 ) + P [b0 , c0 ]+ (c0 , c3 ) + P [c3 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . By the above cases, the lemma is true. Lemma 14. If |Pc | = 1, |P0 | ≤ 2n − 4, |P | ≤ 2n − 2, and u, v are in the same sub-balanced hypercube, then BHn contains a Hamiltonian path passing through P , where n ≥ 2. Proof. The proof is by inductive hypothesis. Assume that BHni −1 is (2n − 4)-prescribed Hamiltonian laceable for i = 0, 1, 2, 3 and n ≥ 2. Since BH1i ∼ = BH1 , by Lemma 4 BH1i is 0-prescribed Hamiltonian laceable for n = 2. By assumption, |Pj | ≤ |P0 | ≤ 2n − 4 for j = 1, 2, 3. By symmetry of BHn , we only need to consider e ∈ E(BHn0−,11 ). Assume that Pc = {e}. Denote e = (a0 , a1 ). We consider the following cases. Case 1. u, v ∈ V (BHn0−1 ). 1,2 2,3 By Lemma 6, there are edges (b1 , b2 ) ∈ E(BHn−1 ), (a2 , a3 ) ∈ E(BHn−1 ). Since |Pi | ≤ 2n − 4 for i = 0, 1, 2, 3, by 0 assumption, BHn−1 contains a Hamiltonian path P0 [u, v] passing through P0 . Assume that (a0 , b0 ) ∈ E(P0 [u, v]) such that 0,1

0,3

j

(a0 , b0 ) ̸ ∈ P0 , (a0 , a1 ) ∈ E(BHn−1 ), and (b0 , b3 ) ∈ E(BHn−1 ). Since |Pj | ≤ 2n − 4 for j = 1, 2, 3 by assumption, BHn−1 contains a Hamiltonian path P [aj , bj ] passing through Pj for j = 1, 2, 3. Let P [u, v] = P0 [u, v] − (a0 , b0 ) + (a0 , a1 ) + P [a1 , b1 ] + (b1 , b2 ) + P [b2 , a2 ] + (a2 , a3 ) + P [a3 , b3 ] + (b3 , b0 ). Hence, P [u, v] is a Hamiltonian path passing through P in BHn . Case 2. u, v ∈ V (BHn1−1 ). Since |P1 | ≤ 2n − 4, by assumption, BHn1−1 contains a Hamiltonian path P1 [u, v] passing through P1 . Assume 0,3 that (a1 , b1 ) ∈ E(P1 [u, v]) such that (a1 , b1 ) ̸ ∈ P1 . By Lemma 6, there are edges (b0 , b3 ) ∈ E(BHn−1 ), (a2 , a3 ) ∈ 2,3

1 ,2

j

E(BHn−1 ), (b1 , b2 ) ∈ E(BHn−1 ). Since |Pj | ≤ 2n − 4 for j = 0, 2, 3, BHn−1 contains a Hamiltonian path P [aj , bj ] passing through Pj for j = 0, 2, 3. Let P [u, v] = P1 [u, v] − (a1 , b1 ) + (a1 , a0 ) + P [a0 , b0 ] + (b0 , b3 ) + P [b3 , a3 ] + (a3 , a2 ) + P [a2 , b2 ] + (b2 , b1 ). Hence, P [u, v] is a Hamiltonian path passing through P in BHn . Case 3. u, v ∈ V (BHn2−1 ). Since |P2 | ≤ 2n − 4, by assumption, BHn2−1 contains a Hamiltonian path P2 [u, v] passing through P2 . Since ℓ(P2 [u, v]) − |P2 | ≥ 22(n−1) − (2n − 4) ≥ 4 for n ≥ 2, we can find at least one edge, denoted by (a2 , b2 ) ∈ E(P2 [u, v]), such that (a2 , b2 ) ̸∈ 1,2 2,3 0,3 P2 . By Definition 2 and Lemma 6, there are edges (b1 , b2 ) ∈ E(BHn−1 ) and (a3 , a2 ) ∈ E(BHn−1 ), (b0 , b3 ) ∈ E(BHn−1 ). Since

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Fig. 7. The illustrations of Cases 1.1, 1.2, and 3 in Lemma 15.

|Pj | ≤ 2n − 4 for j = 0, 1, 3, by assumption, BHnj −1 contains a Hamiltonian path P [aj , bj ] passing through Pj for j = 0, 1, 3. Let P [u, v] = P2 [u, v] − (a2 , b2 ) + (b2 , b1 ) + P [b1 , a1 ] + (a1 , a0 ) + P [a0 , b0 ] + (b0 , b3 ) + P [b3 , a3 ] + (a3 , a2 ). Hence, P [u, v] is a Hamiltonian path passing through P in BHn . Case 4. u, v ∈ V (BHn3−1 ). Since |P3 | ≤ 2n − 4, by assumption, BHn3−1 contains a Hamiltonian path P3 [u, v] passing through P3 . Since ℓ(P3 [u, v]) − |P3 | ≥ 22(n−1) − (2n − 4) ≥ 4 for n ≥ 2, we can find at least one edge, denoted by (a3 , b3 ) ∈ E(P3 [u, v]), such that 1,2 2,3 0 ,3 (a3 , b3 ) ̸ ∈ P3 . By Definition 1 and Lemma 6, there are edges (b0 , b3 ) ∈ E(BHn−1 ), (a2 , a3 ) ∈ E(BHn−1 ), (b1 , b2 ) ∈ E(BHn−1 ). j

j

Since |Pj | ≤ 2n − 4 for j = 0, 1, 2, by assumption, BHn−1 contains a Hamiltonian path P [aj , bj ], BHn−1 for j = 0, 1, 2. Let P [u, v] = P3 [u, v] − (a3 , b3 ) + (b3 , b0 ) + P [b0 , a0 ] + (a0 , a1 ) + P [a1 , b1 ] + (b1 , b2 ) + P [b2 , a2 ] + (a2 , a3 ). Hence, P [u, v] is a Hamiltonian path passing through P in BHn . Lemma 15. If |Pc | = 1, |P0 | ≤ 2n − 4, |P | ≤ 2n − 2, and u, v are in different sub-balanced hypercubes, then BHn contains a Hamiltonian path passing through P , where n ≥ 2.

Proof. The proof is by inductive hypothesis. Assume that BHni −1 is (2n − 4)-prescribed Hamiltonian laceable for i = 0, 1, 2, 3 and n ≥ 2. Since BH1i ∼ = BH1 , by Lemma 4 BH1i is 0-prescribed Hamiltonian laceable for n = 2. Assume that 0,1 Pc = {e}. By symmetry of BHn , we only need to consider e ∈ E(BHn−1 ). Denote e = (a0 , a1 ). By assumption |P0 | = max|{Pi }| for i = 0, 1, 2, 3. So |Pj | ≤ 2n − 4 for j = 1, 2, 3. If |Pi | = 2n − 4 for some i ∈ {1, 2, 3}, then n = 2. (Since |Pi | + |P0 | + |Pc | ≤ |P | ≤ 2n − 2, i.e., (2n − 4) + (2n − 4) + 1 ≤ 2n − 2, we get n ≤ 25 . Since n ≥ 2 and n is an integer, n = 2.) Since BH2 is edge-transitive, we only need to consider e = ((0, 0), (1, 1)). For any two vertices u, v ∈ V (BH2 ), we can find a Hamiltonian path joining u and v passing through e in BH2 . See the Appendix. If |Pi | ≤ 2n − 5 for i ∈ {1, 2, 3}, then n ≥ 3. We consider the following cases. Case 1. u ∈ V (BHn0−1 ). Since e ∈ Pc , |P | − |{e}| ≤ 2n − 3, by Lemma 8, e is contained in 2n − 2 edge-disjoint 8-cycles with only one common edge e, denoted by C8 s, such that |C8 ∩ E(BHni −1 )| = 1 for i = 0, 1, 2, 3. So we can find at least one 8-cycle C ∗ such that |C ∗ ∩ E(BHni −1 )| = 1 for i = 0, 1, 2, 3 and C ∗ ∩ P = e. Denote C ∗ = ⟨a0 , a1 , b1 , b2 , a2 , a3 , b3 , b0 , a0 ⟩, where (ai , bi ) ∈ E(BHni −1 ) for i = 0, 1, 2, 3. Since |{(a0 , b0 )} ∪ P0 | ≤ 2n − 3, by Lemma 10, BHn0−1 contains a Hamiltonian cycle C0 passing through P0 ∪{(a0 , b0 )}. Assume that (u, c0 ) ∈ E(C0 ) such that (u, c0 ) ̸ ∈ P0 . Denote C0 = P [a0 , c0 ]+ (c0 , u) + P [u, b0 ]+ (b0 , a0 ). 0,3 By Definition 2, there is an edge (c0 , c3 ) ∈ E(BHn−1 ). 1 Case 1.1. v ∈ V (BHn−1 ). Since |{(a1 , b1 )} ∪ P1 | ≤ 2n − 3, by Lemma 10, BHn1−1 contains a Hamiltonian cycle C1 passing through P1 ∪ {(a1 , b1 )}. Assume that (v, c1 ) ∈ E(C0 ) such that (v, c1 ) ̸ ∈ P1 . Denote C1 = (a1 , b1 ) + P [b1 , v] + (v, c1 ) + P [c1 , a1 ]. By Definition 2, 1,2 2,3 there is an edge (c1 , c2 ) ∈ E(BHn−1 ). By Lemma 6, there is an edge (d2 , d3 ) ∈ E(BHn−1 ). Since |{(a3 , b3 )}| + |P3 | ≤ 3 1 + (2n − 5) = 2n − 4 for n ≥ 3, by assumption, BHn−1 is (2n − 4)-prescribed Hamiltonian laceable, BHn3−1 contains a Hamiltonian path P [c3 , d3 ] passing through {(a3 , b3 )} ∪ P3 . Since |P2 | + |{(a2 , b2 )}| ≤ (2n − 5) + 1 = 2n − 4, by assumption BHn2−1 is (2n−4)-prescribed Hamiltonian laceable, BHn2−1 contains a Hamiltonian path P [c2 , d2 ] passing through P2 ∪ {(a2 , b2 )}. Denote P [c3 , d3 ] = P [c3 , a3 ] + (a3 , b3 ) + P [b3 , d3 ] and P [c2 , d2 ] = P [c2 , b2 ] + (b2 , a2 ) + P [a2 , d2 ]. Let P [u, v] = P [u, b0 ] + (b0 , b3 ) + P [b3 , d3 ] + (d3 , d2 ) + P [d2 , a2 ] + (a2 , a3 ) + P [a3 , c3 ] + (c3 , c0 ) + P [c0 , a0 ] + (a0 , a1 ) + P [a1 , c1 ] + (c1 , c2 ) + P [c2 , b2 ] + (b2 , b1 ) + P [b1 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . See Fig. 7(a). Case 1.2. v ∈ V (BHn2−1 ). 2,3 By Lemma 6, there is an edge (d2 , d3 ) ∈ E(BHn−1 ). Since |Pj | ≤ 2n − 5 for j = 1, 2, 3 and n ≥ 3, by assumption, 1 BHn−1 contains a Hamiltonian path P [a1 , b1 ] passing through P1 , BHn2−1 contains a Hamiltonian path P [d2 , v] passing through P2 ∪ {(a2 , b2 )}, and BHn3−1 contains a Hamiltonian path P [c3 , d3 ] passing through P3 ∪ {(a3 , b3 )}. Denote P [d2 , v] =

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P [d2 , a2 ] + (a2 , b2 ) + P [b2 , v] and P [c3 , d3 ] = P [c3 , a3 ] + (a3 , b3 ) + P [b3 , d3 ]. Let P [u, v] = P [u, b0 ] + (b0 , b3 ) + P [b3 , d3 ] + (d3 , d2 ) + P [d2 , a2 ] + (a2 , a3 ) + P [a3 , c3 ] + (c3 , c0 ) + P [c0 , a0 ] + (a0 , a1 ) + P [a1 , b1 ] + (b1 , b2 ) + P [b2 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . See Fig. 7(b). Case 1.3. v ∈ V (BHn3−1 ). Since |P1 |+|{(a1 , b1 )}| ≤ 2n − 4 and |P2 | ≤ 2n − 5, by assumption, BHn1−1 contains a Hamiltonian path P [a1 , b1 ] passing through P1 , BHn2−1 contains a Hamiltonian path P [a2 , b2 ] passing through P2 . Since |P3 |+|{(a3 , b3 )}| ≤ (2n − 5) + 1 ≤ 2n − 4 for n ≥ 3, BHn3−1 contains a Hamiltonian path P [c3 , v] passing through P3 ∪{(a3 , b3 )}. Denote P [c3 , v] = P [c3 , b3 ]+(b3 , a3 )+ P [a3 , v]. Let P [u, v] = C0 − (c0 , u) − (a0 , b0 ) + (c0 , c3 ) + P [c3 , b3 ] + (b3 , b0 ) + (a0 , a1 ) + P [a1 , b1 ] + (b1 , b2 ) + P [b2 , a2 ] + (a2 , a3 ) + P [a3 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . Case 2. u ∈ V (BHn1−1 ), v ∈ V (BHn2−1 ). 0,3 2,3 By Lemma 6, there are edges (b0 , b3 ) ∈ E(BHn−1 ), (a3 , a2 ) ∈ E(BHn−1 ). Since |Pi | ≤ 2n − 5 for i = 0, 1, 2, 3 and n ≥ 3, by 1 assumption, BHn−1 contains a Hamiltonian path P [u, a1 ] passing through P1 , BHn0−1 contains a Hamiltonian path P [a0 , b0 ] passing through P0 , BHn3−1 contains a Hamiltonian path P [a3 , b3 ] passing through P3 , and BHn2−1 contains a Hamiltonian path P [v, a2 ] passing through P2 . Let P [u, v] = P [u, a1 ] + (a1 , a0 ) + P [a0 , b0 ] + (b0 , b3 ) + P [b3 , a3 ] + (a3 , a2 ) + P [a2 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . Case 3. u ∈ V (BHn1−1 ), v ∈ V (BHn3−1 ). With the same discussion as Case 1, e is contained in a 8-cycle C ∗ such that |C ∗ ∩ E(BHni −1 )| = 1 and C ∗ ∩ P = {e}. Denote C ∗ = ⟨a0 , a1 , b1 , b2 , a2 , a3 , b3 , b0 , a0 ⟩, where (ai , bi ) ∈ E(BHni −1 ) for i = 0, 1, 2, 3. Since |P0 | ≤ 2n − 4, by Lemma 10, BHn0−1 contains a Hamiltonian cycle C0 passing through P0 ∪ (a0 , b0 ). Since |P1 | + |{(a1 , b1 )}| ≤ (2n − 5) + 1 ≤ 2n − 4 for n ≥ 3, by assumption, BHn1−1 contains a Hamiltonian cycle C1 passing through P1 ∪ {(a1 , b1 )}. Assume that (u, c1 ) ∈ E(C1 ) such that (u, c1 ) ̸ ∈ P1 , where two paths between a1 and u in C1 contains only one vertex of {c1 , b1 } respectively. Denote 0,1 C1 = P [u, b1 ] + (b1 , a1 ) + P [a1 , c1 ] + (c1 , u). Assume that (c1 , c0 ) ∈ E(BHn−1 ) and (c0 , d0 ) ∈ E(C0 ) such that the two paths between a0 and c0 contains only one vertex of b0 and d0 respectively. Denote C0 = P [a0 , d0 ] + (d0 , c0 ) + P [c0 , b0 ] + (b0 , a0 ). 0,3 0,3 1,2 By Definition 2, (d0 , d3 ) ∈ E(BHn−1 ), (b0 , b3 ) ∈ E(BHn−1 ), (b1 , b2 ) ∈ E(BHn−1 ). Since |P2 | ≤ 2n − 5, by assumption, 2 BHn−1 contains a Hamiltonian path P [a2 , b2 ] passing through P2 . Since |P3 | + |{(a3 , b3 )}| ≤ (2n − 5) + 1 ≤ 2n − 4 for n ≥ 3, by assumption, BHn3−1 contains a Hamiltonian path P [d3 , v] passing through P3 ∪ {(a3 , b3 )}. Denote P [d3 , v] = P [d3 , a3 ] + (a3 , b3 ) + P [b3 , v]. Let P [u, v] = P [u, b1 ] + (b1 , b2 ) + P [b2 , a2 ] + (a2 , a3 ) + P [a3 , d3 ] + (d3 , d0 ) + P [d0 , a0 ] + (a0 , a1 ) + P [a1 , c1 ] + (c1 , c0 ) + P [c0 , b0 ] + (b0 , b3 ) + P [b3 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . See Fig. 7(c). Case 4. u ∈ V (BHn2−1 ), v ∈ V (BHn3−1 ). Since |P0 | ≤ 2n − 3, by Lemma 10, BHn0−1 contains a Hamiltonian cycle C0 passing through P0 . Assume that 0,3 0,1 (a0 , b0 ) ∈ E(C0 ) such that (a0 , b0 ) ̸ ∈ P0 . By Definition 2, there are edges (b0 , b3 ) ∈ E(BHn−1 ) and (a0 , a1 ) ∈ E(BHn−1 ). 1,2 By Lemma 6, there is an edge (b1 , b2 ) ∈ E(BHn−1 ). Since |Pj | ≤ 2n − 5 for j = 1, 2, 3 and n ≥ 3, by assumption, BHn1−1 contains a Hamiltonian path P [a1 , b1 ] passing through P1 , BHn2−1 contains a Hamiltonian path P [b2 , u] passing through P2 , and BHn3−1 contains a Hamiltonian path P [b3 , v] passing through P3 . Let P [u, v] = P [u, b2 ] + (b2 , b1 ) + P [b1 , a1 ] + (a1 , a0 ) + C0 − (a0 , b0 ) + (b0 , b3 ) + P [b3 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . Lemma 16. If |Pc | = 1, |P0 | = 2n − 3, |P | = 2n − 2, and u, v are in the same sub-balanced hypercube, then BHn contains a Hamiltonian path P [u, v] passing through P , where n ≥ 2. Proof. The proof is by inductive hypothesis. Assume that BHni −1 is (2n − 4)-prescribed Hamiltonian laceable for i = 0, 1, 2, 3 and n ≥ 2. Since BH1i ∼ = BH1 , by Lemma 4 BH1i is 0-prescribed Hamiltonian laceable for n = 2. Assume that Pc = {e}. We only consider u, v ∈ V (BHn0−1 ), the cases of u, v ∈ V (BHni −1 ) for i = 1, 2, 3 can be considered similarly. In this case, |P1 | = |P2 | = |P3 | = 0. We consider the following cases. Assume that (a0 , b0 ) ∈ P0 . By assumption, BHn0−1 is (2n − 4)-Hamiltonian laceable, BHn0−1 contains a Hamiltonian path P0 [u, v] passing through P0 − (a0 , b0 ). 0,1 3,0 Case 1. e ∈ E(BHn−1 ) or e ∈ E(BHn−1 ). 0,1 By symmetry of BHn , we only need to consider e ∈ E(BHn−1 ). Denote e = (x0 , x1 ). Clearly, x0 ∈ V (P0 [u, v]). Assume that (x0 , y0 ) ∈ E(P0 [u, v]) and (x0 , y0 ) ̸ ∈ P0 . By Definition 2 and Lemma 6, there are 0 ,3 1,2 2,3 edges (y0 , y3 ) ∈ E(BHn−1 ), (y1 , y2 ) ∈ E(BHn−1 ), and (x2 , x3 ) ∈ E(BHn−1 ). Case 1.1. (a0 , b0 ) ̸ ∈ E(P0 [u, v]). Assume that c0 and d0 are neighbors of a0 and b0 , respectively, such that the path between a0 and b0 contains only vertex of {c0 , d0 }. Without loss of generality, we assume that (b0 , d0 ), (a0 , c0 ) ∈ E(P0 [u, v]) and P0 [u, v] is denoted by P0 [u, v] = P [u, d0 ] + (d0 , b0 ) + P [b0 , c0 ] + (c0 , a0 ) + P [a0 , x0 ] + (x0 , y0 ) + P [y0 , v]. By Definition 2, there are edges 0,1 0,3 0,3 1,2 (d0 , d1 ) ∈ E(BHn−1 ), (c0 , c3 ) ∈ E(BHn−1 ), (y0 , y3 ) ∈ E(BHn−1 ). By Lemma 6, there are edges (c1 , c2 ) ∈ E(BHn−1 ), (d2 , d3 ) ∈ 2,3 1 E(BHn−1 ). Since |P1 | = |P2 | = |P3 | = 0, by Lemma 7, BHn−1 contains two vertex-disjoint paths P [x1 , y1 ] and P [c1 , d1 ] such that V (P [x1 , y1 ]) ∪ V (P [c1 , d1 ]) = V (BHn1−1 ), BHn2−1 contains two vertex-disjoint paths P [c2 , d2 ] and P [x2 , y2 ] such that V (P [c2 , d2 ]) ∪ V (P [x2 , y2 ]) = V (BHn2−1 ), and BHn3−1 contains two vertex-disjoint paths P [x3 , y3 ] and P [c3 , d3 ] such that V (P [x3 , y3 ]) ∪ V (P [c3 , d3 ]) = V (BHn3−1 ). Let P [u, v] = P [u, d0 ]+ (d0 , d1 ) + P [d1 , c1 ]+ (c1 , c2 ) + P [c2 , d2 ]+ (d2 , d3 ) + P [d3 , c3 ]+

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Fig. 8. The illustrations of Cases 1.1, 2.1, and 3.1 in Lemma 16.

(c3 , c0 ) + P [c0 , b0 ] + (b0 , a0 ) + P [a0 , x0 ] + (x0 , x1 ) + P [x1 , y1 ] + (y1 , y2 ) + P [y2 , x2 ] + (x2 , x3 ) + P [x3 , y3 ] + (y3 , y0 ) + P [y0 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . See Fig. 8(a). Case 1.2. (a0 , b0 ) ∈ E(P [u, v]). Let P0 [u, v] = P [u, x0 ] + (x0 , y0 ) + P [y0 , v]. By Lemma 4, BHn1−1 contains a Hamiltonian path P [x1 , y1 ], BHn2−1 contains a Hamiltonian path P [x2 , y2 ], and BHn3−1 contains a Hamiltonian path P [x3 , y3 ]. Let P [u, v] = P [u, x0 ] + (x0 , x1 ) + P [x1 , y1 ] + (y1 , y2 ) + P [y2 , x2 ] + (x2 , x3 ) + P [x3 , y3 ] + (y3 , y0 ) + P [y0 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . 1,2 Case 2. e ∈ E(BHn−1 ). Denote e = (x1 , x2 ). Case 2.1. (a0 , b0 ) ̸ ∈ E(P0 [u, v]). Assume that c0 and d0 are neighbors of a0 and b0 , respectively, such that the path between a0 and b0 contains only one vertex of {c0 , d0 }. Denote P0 [u, v] = P [u, a0 ] + (a0 , c0 ) + P [c0 , b0 ] + (b0 , d0 ) + P [d0 , v]. Assume that (d0 , d1 ) ∈ 0 ,3 2,3 0,1 E(BHn−1 ), (a2 , a3 ) ∈ E(BHn−1 ), (c0 , c3 ) ∈ E(BHn−1 ). Since |P1 | = |P2 | = |P3 | = 0, by Lemma 4, BHn1−1 contains a 2 Hamiltonian path P [d1 , x1 ], BHn−1 contains a Hamiltonian path P [x2 , a2 ], and BHn3−1 contains a Hamiltonian path P [a3 , c3 ]. Let P [u, v] = P [u, a0 ]+ (a0 , b0 ) + P [b0 , c0 ]+ (c0 , c3 ) + P [c3 , a3 ]+ (a3 , a2 ) + P [a2 , x2 ]+ (x2 , x1 ) + P [x1 , d1 ]+ (d1 , d0 ) + P [d0 , v]. Hence, P [u, v] is a Hamiltonian path passing through P . See Fig. 8(b). Case 2.2. (a0 , b0 ) ∈ E(P0 [u, v]). Since ℓ(P0 [u, v]) − |P0 | = 22(n−1) − 1 − (2n − 3) ≥ 2 for n ≥ 2. We can find an edge, denoted by (c0 , d0 ) ∈ E(P0 [u, v]), 0,1 such that (c0 , d0 ) ̸ ∈ P0 . Denote P0 [u, v] = P [u, d0 ] + (d0 , c0 ) + P [c0 , v]. By Definition 2, there are edges (c0 , c1 ) ∈ E(BHn−1 ) 0,3 2,3 and (d0 , d3 ) ∈ E(BHn−1 ). By Lemma 6, there is an edge (a2 , a3 ) ∈ E(BHn−1 ). Since |P1 | = |P2 | = |P3 | = 0, by Lemma 4, there are Hamiltonian paths P [c1 , x1 ] in BHn1−1 , P [x2 , a2 ] in BHn2−1 , and P [a3 , d3 ] in BHn3−1 . Let P [u, v] = P [u, d0 ] + (d0 , d3 ) + P [d3 , a3 ] + (a3 , a2 ) + P [a2 , x2 ] + (x2 , x1 ) + P [x1 , c1 ] + (c1 , c0 ) + P [c0 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . 2 ,3 Case 3. e ∈ E(BHn−1 ). Denote e = (x2 , x3 ). Case 3.1. (a0 , b0 ) ̸ ∈ E(P0 [u, v]). Assume that c0 and d0 are neighbors of a0 and b0 , respectively, such that the path between a0 and b0 contains only one vertex of {c0 , d0 }. Denote P0 [u, v] = P [u, a0 ] + (a0 , c0 ) + P [c0 , b0 ] + (b0 , d0 ) + P [d0 , v]. By Definition 1, there are edges 0,1 0,3 1,2 (d0 , d1 ) ∈ E(BHn−1 ) and (c0 , c3 ) ∈ E(BHn−1 ). By Lemma 6, there is an edge (c1 , c2 ) ∈ E(BHn−1 ). Since |P1 | = |P2 | = |P3 | = 0, 1 2 by Lemma 4, BHn−1 contains a Hamiltonian path P [c1 , d1 ], BHn−1 contains a Hamiltonian path P [x2 , c2 ], and BHn3−1 contains a Hamiltonian path P [c3 , x3 ]. Let P [u, v] = P [u, a0 ]+ (a0 , b0 ) + P [b0 , c0 ]+ (c0 , c3 ) + P [c3 , x3 ]+ (x3 , x2 ) + P [x2 , c2 ]+ (c2 , c1 ) + P [c1 , d1 ] + (d1 , d0 ) + P [d0 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . See Fig. 8(c). Case 3.2. (a0 , b0 ) ∈ E(P0 [u, v]). Since ℓ(P0 [u, v])−|P0 | = 22(n−1) −1−(2n−3) ≥ 2 for n ≥ 2. We can find an edge, denoted by, (c0 , d0 ) ∈ E(P0 [u, v]), such 0 ,3 that (c0 , d0 ) ̸ ∈ P0 . Denote P0 [u, v] = P [u, d0 ] + (d0 , c0 ) + P [c0 , v]. By Definition 2, there are edges (d0 , d3 ) ∈ E(BHn−1 ) and 0 ,1 1,2 (c0 , c1 ) ∈ E(BHn−1 ). By Lemma 6, there is an edge (d1 , d2 ) ∈ E(BHn−1 ). Since |P1 | = |P2 | = |P3 | = 0, by Lemma 4, BHn1−1 contains a Hamiltonian path P [c1 , d1 ], BHn2−1 contains a Hamiltonian path P [x2 , d2 ], and BHn3−1 contains a Hamiltonian path P [x3 , d3 ]. Let P [u, v] = P [u, d0 ] + (d0 , d3 ) + P [d3 , x3 ] + (x3 , x2 ) + P [x2 , d2 ] + (d2 , d1 ) + P [d1 , c1 ] + (c1 , c0 ) + P [c0 , v]. Hence, P [u, v] is a Hamiltonian path passing through P . Lemma 17. If |Pc | = 1, |P0 | = 2n − 3, |P | = 2n − 2, and u, v are in different sub-balanced hypercubes, then BHn contains a Hamiltonian path P [u, v] passing through P , where n ≥ 2.

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Fig. 9. The illustrations Cases 1.1.1, 1.1.2 and 1.3 in Lemma 17.

Proof. The proof is by inductive hypothesis. Assume that BHni −1 is (2n − 4)-prescribed Hamiltonian laceable for i = 0, 1, 2, 3 and n ≥ 2. Since BH1i ∼ = BH1 , by Lemma 4 BH1i is 0-prescribed Hamiltonian laceable for n = 2. Clearly, |P1 | = |P2 | = |P3 | = 0. Denote Pc = {e}. 0 ,1 0 ,3 Case 1. e ∈ E(BHn−1 ) or e ∈ E(BHn−1 ). 0 ,1 By symmetry of BHn , we only need to consider e ∈ E(BHn−1 ). Denote e = (x0 , x1 ). Since |P0 | = 2n − 3, by Lemma 10, 0 BHn−1 contains a Hamiltonian cycle C0 passing through P0 . Assume that (x0 , y0 ) ∈ E(C0 ) such that (x0 , y0 ) ̸ ∈ P0 . By Definition 2, there is an edge (y0 , y3 ) ∈ E(BHn3−1 ). We consider the following cases. Case 1.1. u ∈ V (BHn0−1 ). Assume that (u, a0 ) ∈ E(C0 ), (x0 , y0 ) ∈ E(C0 ) such that one path between u and x0 contains two vertices of {a0 , y0 }, and the other path between u and x0 does not contain vertices of {a0 , y0 }. Denote C0 = (u, a0 ) + P [a0 , y0 ] + (y0 , x0 ) + P [x0 , u]. 0 ,3 0 ,3 By Definition 1, there are edges (a0 , a3 ) ∈ E(BHn−1 ), (y0 , y3 ) ∈ E(BHn−1 ). We consider the following cases. 1 Case 1.1.1. v ∈ V (BHn−1 ). 2,3 2,3 1,2 1,2 By Lemma 6, there are edges (y1 , y2 ) ∈ E(BHn−1 ), (a1 , a2 ) ∈ E(BHn−1 ), (x2 , x3 ) ∈ E(BHn−1 ), (b2 , b3 ) ∈ E(BHn−1 ). Since |P1 | = |P2 | = |P3 | = 0, by Lemma 7, BHn1−1 contains two vertex-disjoint paths P [x1 , y1 ] and P [v, a1 ] such that V (P [x1 , y1 ]) ∪ V (P [v, a1 ]) = V (BHn1−1 ), BHn2−1 contains two vertex-disjoint paths P [x2 , y2 ] and P [a2 , b2 ] such that V (P [x2 , y2 ]) ∪ V (P [a2 , b2 ]) = V (BHn2−1 ), and BHn3−1 contains two vertex-disjoint paths P [a3 , b3 ] and P [x3 , y3 ] such that V (P [a3 , b3 ]) ∪ V (P [x3 , y3 ]) = V (BHn3−1 ). Let P [u, v] = P [u, x0 ]+ (x0 , x1 ) + P [x1 , y1 ]+ (y1 , y2 ) + P [y2 , x2 ]+ (x2 , x3 ) + P [x3 , y3 ]+ (y3 , y0 ) + P [y0 , a0 ] + (a0 , a3 ) + P [a3 , b3 ] + (b3 , b2 ) + P [b2 , a2 ] + (a2 , a1 ) + P [a1 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . See Fig. 9(a). Case 1.1.2. v ∈ V (BHn2−1 ). 1,2 2,3 2 ,3 By Lemma 6, there are edges (y1 , y2 ) ∈ E(BHn−1 ), (x2 , x3 ) ∈ E(BHn−1 ), (b2 , b3 ) ∈ E(BHn−1 ). Since |P1 | = 0, by Lemma 4, 1 2 BHn−1 contains a Hamiltonian path P [x1 , y1 ]. Since |P2 | = |P3 | = 0, by Lemma 7, BHn−1 contains two vertex-disjoint paths P [x2 , y2 ] and P [b2 , v] such that V (P [x2 , y2 ]) ∪ V (P [b2 , v]) = V (BHn2−1 ), BHn3−1 contains two vertex-disjoint paths P [a3 , b3 ] and P [x3 , y3 ] such that V (P [a3 , b3 ]) ∪ V (P [x3 , y3 ]) = V (BHn3−1 ). Let P [u, v] = P [u, x0 ] + (x0 , x1 ) + P [x1 , y1 ] + (y1 , y2 ) + P [y2 , x2 ] + (x2 , x3 ) + P [x3 , y3 ] + (y3 , y0 ) + P [y0 , a0 ] + (a0 , a3 ) + P [a3 , b3 ] + (b3 , b2 ) + P [b2 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . See Fig. 9(b). Case 1.1.3. v ∈ V (BHn3−1 ). 2,3 1,2 By Lemma 6, there are edges (x3 , x2 ) ∈ E(BHn−1 ), (y1 , y2 ) ∈ E(BHn−1 ). Since |P1 | = |P2 | = 0, by Lemma 4, BHn1−1 2 contains a Hamiltonian path P [x1 , y1 ] and BHn−1 contains a Hamiltonian path P [x2 , y2 ]. Since |P3 | = 0, by Lemma 7, BHn3−1 contains two vertex-disjoint paths P [a3 , v] and P [x3 , y3 ] such that V (P [a3 , v]) ∪ V (P [x3 , y3 ]) = V (BHn3−1 ). Let P [u, v] = P [u, x0 ] + (x0 , x1 ) + P [x1 , y1 ] + (y1 , y2 ) + P [y2 , x2 ] + (x2 , x3 ) + P [x3 , y3 ] + (y3 , y0 ) + P [y0 , a0 ] + (a0 , a3 ) + P [a3 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . Case 1.2. u ∈ V (BHn1−1 ), v ∈ V (BHn2−1 ). 2,3 By Lemma 6, there is an edge (x2 , x3 ) ∈ E(BHn−1 ). Since |P1 | = |P2 | = |P3 | = 0, by Lemma 4, BHn1−1 contains a 2 Hamiltonian path P [x1 , u], BHn−1 contains a Hamiltonian path P [x2 , v], and BHn3−1 contains a Hamiltonian path P [x3 , y3 ]. Let P [u, v] = P [u, x1 ] + (x1 , x0 ) + C0 − (x0 , y0 ) + (y0 , y3 ) + P [y3 , x3 ] + (x3 , x2 ) + P [x2 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . Case 1.3. u ∈ V (BHn1−1 ), v ∈ V (BHn3−1 ). Since ℓ(C0 ) − |{(x0 , y0 )}| − |P0 | = 22(n−1) − 1 − (2n − 3) ≥ 2 for n ≥ 2, we can find an edge, denoted by (a0 , b0 ) ∈ E(C0 ), such that (a0 , b0 ) ̸ ∈ P0 . Denote C0 = P [x0 , a0 ] + (a0 , b0 ) + P [b0 , y0 ] + (y0 , x0 ). By Definition 1, there 0,1 0,3 0,3 are edges (a0 , a1 ) ∈ E(BHn−1 ), (b0 , b3 ) ∈ E(BHn−1 ), and (y0 , y3 ) ∈ E(BHn−1 ). Since |P1 | = 0, by Lemma 4, BHn1−1 contains 1,2 a Hamiltonian path P [u, x1 ]. Clearly, a1 ∈ V (P [u, x1 ]). Assume that (a1 , b1 ) ∈ E(P [x1 , u]) and (b1 , b2 ) ∈ E(BHn−1 ). Denote

D. Cheng / Discrete Applied Mathematics 262 (2019) 56–71

69 2,3

P [u, x1 ] = P [u, a1 ] + (a1 , b1 ) + P [b1 , x1 ]. By Lemma 6, there is an edge (a2 , a3 ) ∈ E(BHn−1 ). By Lemma 4, BHn2−1 contains a Hamiltonian path P [a2 , b2 ]. Since |P3 | = 0, by Lemma 7, BHn3−1 contains two vertex-disjoint paths P [y3 , v] and P [a3 , b3 ] such that V (P [y3 , v]) ∪ V (P [a3 , b3 ]) = V (BHn3−1 ). Let P [u, v] = P [u, a1 ] + (a1 , a0 ) + P [a0 , x0 ] + (x0 , x1 ) + P [x1 , b1 ] + (b1 , b2 ) + P [b2 , a2 ]+ (a2 , a3 ) + P [a3 , b3 ]+ (b3 , b0 ) + P [b0 , y0 ]+ (y0 , y3 ) + P [y3 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . See Fig. 9(c). Case 1.4. u ∈ V (BHn2−1 ), v ∈ V (BHn3−1 ). 1 ,2 By Lemma 6, there is an edge (y1 , y2 ) ∈ E(BHn−1 ). Since |P1 | = |P2 | = |P3 | = 0, by Lemma 4, BHn2−1 contains a 2 Hamiltonian path P [x1 , y1 ], BHn−1 contains a Hamiltonian path P [y2 , u], and BHn3−1 contains a Hamiltonian path P [y3 , v]. Let P [u, v] = P [u, y2 ] + (y2 , y1 ) + P [y1 , x1 ] + (x1 , x0 ) + C0 − (x0 , y0 ) + (y0 , y3 ) + P [y3 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . 1 ,2 2,3 Case 2. e ∈ E(BHn−1 ) or e ∈ E(BHn−1 ). 1,2 By symmetry of BHn , we only need to consider e ∈ E(BHn−1 ). Denote e = (x1 , x2 ). 0 Case 2.1. u ∈ V (BHn−1 ). Since |P0 | = 2n − 3, by Lemma 10, BHn0−1 contains a Hamiltonian cycle C0 passing through P0 . Since ℓ(C0 ) − |P0 | = 2(n−1) 2 − (2n − 3) ≥ 3 for n ≥ 2, we can find at least two vertex-disjoint edges, denoted by (u, a0 ) ∈ E(C0 ), (c0 , d0 ) ∈ E(C0 ), such that (u, a0 ) ̸ ∈ P0 , (c0 , d0 ) ̸ ∈ P0 . Denote C0 = P [u, c0 ] + (c0 , d0 ) + P [d0 , a0 ] + (a0 , u). By Definition 2, there are edges 0 ,1 0,3 0,3 (c0 , c1 ) ∈ E(BHn−1 ), (a0 , a3 ) ∈ E(BHn−1 ), (d0 , d3 ) ∈ E(BHn−1 ). 1 Case 2.1.1. v ∈ V (BHn−1 ). 2 ,3 2,3 1 ,2 By Lemma 6, there are edges (y1 , y2 ) ∈ E(BHn−1 ), (c2 , c3 ) ∈ E(BHn−1 ), (b2 , b3 ) ∈ E(BHn−1 ). Since |P1 | = |P2 | = |P3 | = 0, 1 by Lemma 7, BHn−1 contains two vertex-disjoint paths P [c1 , x1 ] and P [y1 , v] such that V (P [c1 , x1 ])∪V (P [y1 , v]) = V (BHn1−1 ), BHn2−1 contains two vertex-disjoint paths P [c2 , x2 ] and P [b2 , y2 ] such that V (P [c2 , x2 ]) ∪ V (P [b2 , y2 ]) = V (BHn2−1 ), and BHn3−1 contains two vertex-disjoint paths P [c3 , d3 ] and P [a3 , b3 ] such that V (P [c3 , d3 ]) ∪ V (P [a3 , b3 ]) = V (BHn3−1 ). Let P [u, v] = P [u, c0 ] + (c0 , c1 ) + P [c1 , x1 ] + (x1 , x2 ) + P [x2 , c2 ] + (c2 , c3 ) + P [c3 , d3 ] + (d3 , d0 ) + P [d0 , a0 ] + (a0 , a3 ) + P [a3 , b3 ] + (b3 , b2 ) + P [b2 , y2 ] + (y2 , y1 ) + P [y1 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . Case 2.1.2. v ∈ V (BHn2−1 ). 2 ,3 2,3 By Lemma 6, there are edges (c2 , c3 ) ∈ E(BHn−1 ), (b2 , b3 ) ∈ E(BHn−1 ). Since |P1 | = 0, by Lemma 4, BHn1−1 contains a 2 Hamiltonian path P [c1 , x1 ]. Since |P2 | = |P3 | = 0, by Lemma 7, BHn−1 contains two vertex-disjoint paths P [c2 , x2 ] and P [b2 , v] such that V (P [c2 , x2 ]) ∪ V (P [b2 , v]) = V (BHn2−1 ), BHn3−1 contains two vertex-disjoint paths P [c3 , d3 ] and P [a3 , b3 ] such that V (P [a3 , b3 ]) ∪ V (P [c3 , d3 ]) = V (BHn3−1 ). Let P [u, v] = P [u, c0 ] + (c0 , c1 ) + P [c1 , x1 ] + (x1 , x2 ) + P [x2 , c2 ] + (c2 , c3 ) + P [c3 , d3 ] + (d3 , d0 ) + P [d0 , a0 ] + (a0 , a3 ) + P [a3 , b3 ] + (b3 , b2 ) + P [b2 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . Case 2.1.3. v ∈ V (BHn3−1 ). 2,3 By Lemma 6, there is an edge (y2 , y3 ) ∈ E(BHn−1 ). Since |P1 | = |P2 | = 0, by Lemma 4, BHn1−1 contains a Hamiltonian 2 path P [c1 , x1 ], BHn−1 contains a Hamiltonian path P [x2 , y2 ]. Since |P3 | = 0, by Lemma 7, BHn3−1 contains two vertex-disjoint paths P [d3 , v] and P [a3 , y3 ] such that V (P [d3 , v]) ∪ V (P [a3 , y3 ]) = V (BHn3−1 ). Let P [u, v] = P [u, c0 ] + (c0 , c1 ) + P [c1 , x1 ] + (x1 , x2 ) + P [x2 , y2 ] + (y2 , y3 ) + P [y3 , a3 ] + (a3 , a0 ) + P [a0 , d0 ] + (d0 , d3 ) + P [d3 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . Case 2.2. u ∈ V (BHn1−1 ). Since |P0 | = 2n − 3, by Lemma 10, BHn0−1 contains a Hamiltonian cycle C0 passing through P0 . Since ℓ(C0 ) − |P0 | = 2(n−1) 2 − (2n − 3) ≥ 3 for n ≥ 2, we can find at least two vertex-disjoint edges, denoted by (a0 , b0 ) ∈ E(C0 ), (c0 , d0 ) ∈ E(C0 ), such that (a0 , b0 ) ̸ ∈ P0 , (c0 , d0 ) ̸ ∈ P0 . Denote C0 = P [a0 , c0 ] + (c0 , d0 ) + P [d0 , b0 ] + (b0 , a0 ). By Definition 2, there are 0,1 0,1 0,3 0 ,3 edges (a0 , a1 ) ∈ E(BHn−1 ), (c0 , c1 ) ∈ E(BHn−1 ), (b0 , b3 ) ∈ E(BHn−1 ), (d0 , d3 ) ∈ E(BHn−1 ). By Lemma 6, there are edges 2,3 2,3 (c2 , c3 ) ∈ E(BHn−1 ), (a2 , a3 ) ∈ E(BHn−1 ). Case 2.2.1. v ∈ V (BHn2−1 ). Since |P1 | = |P2 | = |P3 | = 0, by Lemma 7, BHn1−1 contains two vertex-disjoint paths P [u, a1 ] and P [c1 , x1 ] such that V (P [u, a1 ]) ∪ V (P [c1 , x1 ]) = V (BHn1−1 ), BHn2−1 contains two vertex-disjoint paths P [x2 , a2 ] and P [c2 , v] such that V (P [x2 , a2 ]) ∪ V (P [c2 , v]) = V (BHn2−1 ), and BHn3−1 contains two vertex-disjoint paths P [d3 , c3 ] and P [b3 , a3 ] such that V (P [d3 , c3 ]) ∪ V (P [b3 , a3 ]) = V (BHn3−1 ). Let P [u, v] = P [u, a1 ] + (a1 , a0 ) + P [a0 , c0 ] + (c0 , c1 ) + P [c1 , x1 ] + (x1 , x2 ) + P [x2 , a2 ] + (a2 , a3 ) + P [a3 , b3 ] + (b3 , b0 ) + P [b0 , d0 ] + (d0 , d3 ) + P [d3 , c3 ] + (c3 , c2 ) + P [c2 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . Case 2.2.2. v ∈ V (BHn3−1 ). 2,3 By Lemma 6, there is an edge (c2 , c3 ) ∈ E(BHn−1 ). Since |P2 | = 0, by Lemma 4, BHn2−1 contains a Hamiltonian path P [x2 , c2 ]. Since |P1 | = |P3 | = 0, by Lemma 7, BHn1−1 contains two vertex-disjoint paths P [c1 , x1 ] and P [a1 , u] such that V (P [c1 , x1 ]) ∪ V (P [a1 , u]) = V (BHn1−1 ), BHn3−1 contains two vertex-disjoint paths V (P [b3 , v]) ∪ V (P [c3 , d3 ]) = V (BHn3−1 ). Let P [u, v] = P [u, a1 ] + (a1 , a0 ) + P [a0 , c0 ] + (c0 , c1 ) + P [c1 , x1 ] + (x1 , x2 ) + P [x2 , c2 ] + (c2 , c3 ) + P [c3 , d3 ] + (d3 , d0 ) + P [d0 , b0 ] + (b0 , b3 ) + P [b3 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . Case 2.3. u ∈ V (BHn2−1 ), v ∈ V (BHn3−1 ). By Lemma 10, BHn0−1 contains a Hamiltonian cycle C0 passing through P0 . Since ℓ(C0 ) − |P0 | = 22(n−1) − (2n − 3) ≥ 3 for n ≥ 2, we can find at least one edge, denoted by (a0 , b0 ) ∈ E(C0 ), such that (a0 , b0 ) ̸ ∈ P0 . By Definition 2, there

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Table 1 Hamiltonian paths P [u, v]s pass through e.

u

v

Hamiltonian paths P [u, v]s pass through e.

(2, 0) (2, 0) (2, 0) (2, 0) (2, 0) (0, 1) (0, 1) (0, 1) (0, 1) (2, 1) (2, 1) (2, 1) (2, 1) (0, 2) (0, 2) (2, 2) (2, 2)

(3, 1) (1, 2) (3, 2) (1, 3) (3, 3) (1, 2) (3, 2) (1, 3) (3, 3) (1, 2) (3, 2) (1, 3) (3, 3) (1, 3) (3, 3) (1, 3) (3, 3)

⟨(2, 0), (1, 0), (0, 3), (3, 0), (0, 0), (1, 1), (2, 1), (3, 2), (2, 2), (3, 3), (2, 3), (1, 3), (0, 2), (1, 2), (0, 1), (3, 1)⟩ ⟨(2, 0), (1, 0), (0, 3), (3, 0), (0, 0), (1, 1), (0, 1), (3, 1), (2, 1), (3, 2), (0, 2), (1, 3), (2, 3), (3, 3), (2, 2), (1, 2)⟩ ⟨(2, 0), (1, 0), (0, 3), (3, 0), (0, 0), (1, 1), (0, 1), (3, 1), (2, 1), (1, 2), (2, 2), (3, 3), (2, 3), (1, 3), (0, 2), (3, 2)⟩ ⟨(2, 0), (1, 0), (0, 3), (3, 0), (0, 0), (1, 1), (0, 1), (3, 1), (2, 1), (1, 2), (2, 2), (3, 2), (0, 2), (3, 3), (2, 3), (1, 3)⟩ ⟨(2, 0), (1, 0), (0, 3), (3, 0), (0, 0), (1, 1), (0, 1), (3, 1), (2, 1), (1, 2), (2, 2), (3, 2), (0, 2), (1, 3), (2, 3), (3, 3)⟩ ⟨(0, 1), (3, 1), (2, 0), (1, 1), (0, 0), (1, 0), (0, 3), (3, 0), (2, 3), (1, 3), (0, 2), (3, 3), (2, 2), (3, 2), (2, 1), (1, 2)⟩ ⟨(0, 1), (3, 1), (2, 0), (1, 1), (0, 0), (1, 0), (0, 3), (3, 0), (2, 3), (1, 3), (0, 2), (3, 3), (2, 2), (1, 2), (2, 1), (3, 2)⟩ ⟨(0, 1), (3, 1), (2, 0), (1, 1), (0, 0), (1, 0), (0, 3), (3, 0), (2, 3), (3, 3), (0, 2), (3, 2), (2, 1), (1, 2), (2, 2), (1, 3)⟩ ⟨(0, 1), (3, 1), (2, 0), (1, 1), (0, 0), (1, 0), (0, 3), (3, 0), (2, 3), (1, 3), (0, 2), (3, 2), (2, 1), (1, 2), (2, 2), (3, 3)⟩ ⟨(2, 1), (3, 1), (0, 1), (1, 1), (0, 0), (1, 0), (2, 0), (3, 0), (0, 3), (1, 3), (2, 3), (3, 3), (2, 2), (3, 2), (0, 2), (1, 2)⟩ ⟨(2, 1), (3, 1), (0, 1), (1, 1), (0, 0), (1, 0), (2, 0), (3, 0), (0, 3), (1, 3), (2, 3), (3, 3), (2, 2), (1, 2), (0, 2), (3, 2)⟩ ⟨(2, 1), (3, 2), (0, 1), (3, 1), (2, 0), (1, 1), (0, 0), (1, 0), (0, 3), (3, 0), (2, 3), (3, 3), (0, 2), (1, 2), (2, 2), (1, 3)⟩ ⟨(2, 1), (3, 2), (0, 1), (3, 1), (2, 0), (1, 1), (0, 0), (1, 0), (0, 3), (3, 0), (2, 3), (1, 3), (0, 2), (1, 2), (2, 2), (3, 3)⟩ ⟨(0, 2), (1, 2), (2, 2), (3, 2), (0, 1), (3, 1), (2, 1), (1, 1), (0, 0), (1, 0), (2, 0), (3, 0), (0, 3), (3, 3), (2, 3), (1, 3)⟩ ⟨(0, 2), (1, 2), (2, 2), (3, 2), (0, 1), (3, 1), (2, 1), (1, 1), (0, 0), (1, 0), (2, 0), (3, 0), (0, 3), (1, 3), (2, 3), (3, 3)⟩ ⟨(2, 2), (3, 2), (0, 2), (1, 2), (2, 1), (3, 1), (0, 1), (1, 1), (0, 0), (1, 0), (2, 0), (3, 0), (0, 3), (3, 3), (2, 3), (1, 3)⟩ ⟨(2, 2), (3, 2), (0, 2), (1, 2), (2, 1), (3, 1), (0, 1), (1, 1), (0, 0), (1, 0), (2, 0), (3, 0), (0, 3), (1, 3), (2, 3), (3, 3)⟩

0,1

0 ,3

are edges (a0 , a1 ) ∈ E(BHn−1 ), (b0 , b3 ) ∈ E(BHn−1 ). Since |P1 | = |P2 | = |P3 | = 0, by Lemma 4, BHn1−1 contains a Hamiltonian path P [a1 , x1 ], BHn2−1 contains a Hamiltonian path P [u, x2 ], and BHn3−1 contains a Hamiltonian path P [b3 , v]. Let P [u, v] = P [u, x2 ] + (x2 , x1 ) + P [x1 , a1 ] + (a1 , a0 ) + C0 − (a0 , b0 ) + (b0 , b3 ) + P [b3 , v]. Hence, P [u, v] is a Hamiltonian path passing through P in BHn . Theorem 1. Let P be a set of at most 2n − 2 edges of BHn and u, v be any two vertices from different partite sets of V (BHn ) and P ∪ {u, v} are compatible, then BHn has a Hamiltonian path P [u, v] joining u and v passing through P if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths, where n ≥ 1. Proof. The proof of this theorem is by induction hypothesis. Since BH1 is a 4-cycle, the theorem is true for n = 1. Assume that the theorem is true for k < n, we consider k = n and n ≥ 2 as follows. By Lemma 9, BHn can be partitioned into four sub-balanced hypercubes, denoted by BHn0−1 , BHn1−1 , BHn2−1 , BHn3−1 , such that |P ∩ Ec | ≤ 1. By Lemmas 12–17, the theorem is true. By Theorem 1, we directly obtain the following corollary. Corollary 1. Let P be a set of edges of BHn with |P | ≤ 2n − 1. Then BHn has a Hamiltonian cycle of BHn passing through P if and only if the subgraph induced by P consists of vertex-disjoint paths, where n ≥ 1. 4. Concluding remarks Let P be a set of edges in BHn . In this paper, we prove that for any two vertices u and v from different partite sets of BHn and {u, v} and P are compatible, then BHn contains a Hamiltonian path P [u, v] passing through P with |P | ≤ 2n − 2 if and only if the subgraph inducted by P consists of vertex-disjoint paths. By this result, we directly obtain that BHn contains a Hamiltonian cycle passing through a set of edges of P with |P | ≤ 2n − 1 if and only if the subgraph induced by P consists of vertex-disjoint paths. Acknowledgments The author would like to thank anonymous referees for their valuable suggestions and comments. This work was supported by Tianyuan Fund of National Natural Science Foundation of China [Grant Number 11626114], National Natural Science Foundation of China (Grant Number 11701218), and the PhD Start-up Fund of Natural Science Foundation of Guangdong Province, China (Grant Number 2016A030310082). Appendix Proof. The basic case of Lemma 15. By Lemma 3, we may assume that e = ((0, 0), (1, 1)). Hamiltonian paths P [u, v]s join u and v passing through e are listed in Table 1.

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