Cycles embedding in folded hypercubes under the conditional fault model

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Cycles embedding in folded hypercubes under the conditional fault model Dongqin Cheng Department of Mathematics, Jinan University, 510632, Guangzhou, China

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Article history: Received 25 April 2016 Received in revised form 23 November 2016 Accepted 20 February 2017 Available online xxxx Keywords: Interconnection network Folded hypercube Cycle embedding Fault-tolerant Conditional fault model

a b s t r a c t A faulty network G is under the conditional fault model, i.e., every fault-free vertex of G is incident to at least two fault-free edges. Let FFv and FFe be the set of faulty vertices and faulty edges in FQn , respectively. In this paper, we consider FQn under the conditional fault model and prove that if |FFv | + |FFe | ≤ 2n − 4 and n ≥ 3, then FQn − FFv − FFe contains a fault-free cycle of every even length from 4 to 2n − 2|FFv |; if |FFv | + |FFe | ≤ 2n − 5 and n ≥ 4 is even, then FQn − FFv − FFe contains a fault-free cycle of every odd length from n + 1 to 2n − 2|FFv | − 1. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Interconnection networks are important to parallel computing systems [22], because their processors are connected according to a given interconnection network. Interconnection network topology is usually represented by a simple graph G = (V (G), E(G)), where vertices and edges represent processors and links between processors, respectively. Various kinds of interconnection network topologies have been proposed, such as hypercubes [4], crossed cube [8], folded hypercube [9], twisted cube [10], locally twisted cube [34], and so on. Among them, one famous interconnection network is folded hypercube [9], which is constructed from hypercube [4] by adding edges between any two vertices with complementary addresses. The folded hypercubes have been shown to possess more superior properties than hypercubes [9,30]. When evaluate an interconnection network, one important aspect is its graph embedding capabilities. An embedding between two graphs G (called guest graph) and H (called host graph) is a one-to-one mapping f from V (G) to V (H), where f maps an edge of G to a path of H [22]. Paths and cycles, due to their simple structures, are fundamental topological for parallel and distributed processing [31] and have a wide applications, readers may find details from [1,22]. An intuitive idea is embedding cycles in interconnection networks. Cycles of all kinds of lengths embedding in various interconnection networks have attracted many scholars’ attention in recent years, such as [5,11,12,15,16,18–20,23,27–30,32,33,35]. Since processor or link may fail when an interconnection network is applied to the real world, we need to consider interconnection networks with faulty elements (i.e., faulty edges and/or faulty vertices). Let F be a set of faulty edges and nodes in a graph G. A vertex is fault-free if it is not in F , and an edge is fault-free if its two end-vertices as well as the links between them are not in F . When embedding fault-free cycles in a faulty interconnection network, there are two models to consider, one is standard fault model and the other is conditional fault model [12]. There is no restrictions on the distributions of faulty elements for the former model, but for the latter model we require that every fault-free vertex is incident to at least two fault-free edges [12]. Throughout this paper, let FFe and FFv be the set of faulty edges and faulty vertices in FQn , respectively. Kuo [18] considered FQn with |FFe | + |FFv | ≤ n − 1 and proved that FQn − FFe − FFv contains a fault-free cycle of every even length ℓ E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.dam.2017.02.020 0166-218X/© 2017 Elsevier B.V. All rights reserved.

Please cite this article in press as: D. Cheng, Cycles embedding in folded hypercubes under the conditional fault model, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.02.020.

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with 4 ≤ ℓ ≤ 2n − 2|FFv |, where n ≥ 3; and FQn − FFe − FFv contains a fault-free cycle of every odd length ℓ′ with n + 1 ≤ ℓ′ ≤ 2n − 2|FFv | − 1, where n ≥ 4 and n is even. Fu [11] proved that FQn − FFe − FFv contains a fault-free cycle of length ℓ with ℓ ≥ 2n − 2|FFv | if |FFe | + |FFv | ≤ 2n − 4 and |FFe | ≤ n − 1, where n ≥ 3. Hsieh et al. [14] further considered |FFv | + |FFe | ≤ 2n − 4 and |FFe | ≥ n and obtained the same result as Fu [11]. Cheng et al. [5] considered FQn with |FFv | ≤ n − 2 and showed that if n ≥ 3, then every edge of FQn − FFv lies on a fault-free cycle of every even length ℓ with 4 ≤ ℓ ≤ 2n − 2|FFv |, and if n ≥ 2 and n is even, then every edge of FQn − FFv lies on a fault-free cycle of every odd length ℓ′ with n + 1 ≤ ℓ′ ≤ 2n − 2|FFv | − 1. Kuo and Hsieh [20] proved that when FQn is under the conditional fault model with |FFe | ≤ 2n − 3, FQn − FFe contains a cycle of every even length ℓ with 4 ≤ ℓ ≤ 2n ; if n ≥ 2 being even, FQn − FFe contains a cycle of every odd length ℓ′ with n + 1 ≤ ℓ′ ≤ 2n − 1. Cheng et al. [7] considered FQn with |FFe | ≤ 2n − 4 under the conditional fault model, and proved that every edge of FQn − FFe lies on a fault-free cycle of every even length ℓ with 6 ≤ ℓ ≤ 2n , where n ≥ 5. Cheng et al. [6] also considered odd cycle embedding in FQn with |FFe | ≤ 2n − 5 under the conditional fault model, and proved that every fault-free edge of FQn lies on a fault-free cycle of every odd length ℓ′ with n + 1 ≤ ℓ′ ≤ 2n − 1, where n ≥ 4 and n is even. Kuo [19] proved that under the conditions that (i) each vertex is incident to at least 4 fault-free neighbors; (ii) the number of faulty vertices which are incident to the end-vertices of any fault-free edge e ∈ E(FQn ) is no more than n − 3, then (1) for n ≥ 4, FQn − FFv contains a fault-free cycle of every even length ℓ with 4 ≤ ℓ ≤ 2n − 2|FFv |, where |FFv | ≤ 2n − 7; (2) for n ≥ 4 being even, FQn − FFv contains a fault-free cycle of every odd length ℓ′ with n + 1 ≤ ℓ′ ≤ 2n − 2|FFv | − 1, where |FFv | ≤ 2n − 7. Recently, Kuo and Stewart [21] proved that in FQn with |FFv | + |FFe | ≤ n − 2 and n ≥ 2 is even, any fault-free edge e is contained in a fault-free cycle of odd length ℓ with n + 1 ≤ ℓ ≤ 2n − 2|FFv | − 1. In this paper, we consider FQn under the conditional fault model and prove that in FQn with |FFe |+|FFv | ≤ 2n−4 and n ≥ 3, FQn − FFv − FFe contains a fault-free cycle of every even length ℓ with 4 ≤ ℓ ≤ 2n − 2|FFv |, in FQn with |FFe | + |FFv | ≤ 2n − 5 and n ≥ 4 is even, FQn − FFv − FFe contains a fault-free cycle of every odd length ℓ with n + 1 ≤ ℓ ≤ 2n − 2|FFv | − 1. This paper consists of five sections. In the next section, we present some definitions of graphs, hypercubes, folded hypercubes, as well as some lemmas of hypercubes and folded hypercubes. Our main results are presented in Sections 3 and 4, but the proof of the basic case of Theorem 1 is in Appendix. Finally, concluding remarks are given in Section 5. 2. Preliminaries A graph G = (V , E) is an ordered pair, where V called vertex set is a finite set, and E called edge set is a subset of {(u, v )|(u, v ) is an unordered pair of V }. If (x, y) ∈ E, then x and y are adjacent. A graph G = (V , E) is bipartite if V can be partitioned into two parts such that each edge with two end-vertices from different parts. A path between v0 and vk , denoted by P [v0 , vk ] = ⟨v0 , v1 , . . . , vk ⟩, is a finite sequence of adjacent vertices, where all the vertices are different and v0 and vk are called end-vertices. A cycle is a path of length at least three with the same end-vertices. An edge can be regarded as a special path of length 1. The length of a path P (respectively, a cycle C ), denoted by ℓ(P) (respectively, ℓ(C )), is the number of edges in P (respectively, C ). Two edges are adjacent (respectively, disjoint) if they have one (respectively, no) common end-vertex. Two paths are vertex-disjoint (disjoint, for short) (respectively, edge-disjoint) if they do not have common vertices (respectively, edges). A cycle (respectively, path) of length ℓ is denoted by ℓ-cycle (respectively, ℓ-path). The distance from u to v , denoted by dG (u, v ), is the shortest path between u an v . A graph G is pancyclic if G contains cycles of every length ℓ with 3 ≤ ℓ ≤ |V (G)| [3]. A graph is panconnected if there exists a path of length ℓ with dG (u, v ) ≤ ℓ ≤ |V (G)| − 1 between any two different vertices u and v [2]. A bipartite graph possesses similar properties but the lengths of the cycles are even. A bipartite graph is bipancyclic if G contains cycles of every even length ℓ with 4 ≤ ℓ ≤ |V (G)| [25]. A bipartite graph is bipanconnected if there exists a path of length ℓ with dG (u, v ) ≤ ℓ ≤ |V (G)| − 1 and 2 |(ℓ − dG (u, v )) between any two different vertices u and v [17]. An n-dimensional hypercube is denoted by Qn = (V (Qn ), E(Qn )), where V (Qn ) = {x = x1 x2 . . . xn |xi ∈ {0, 1} for 1 ≤ i ≤ n} and E(Qn ) = {(x, x(k) )|x = x1 x2 . . . xk−1 xk xk+1 . . . xn ∈ V (Qn ) and x(k) = x1 x2 . . . xk−1 xk xk+1 . . . xn , xk = 1 − xk , 1 ≤ k ≤ n}. (x, x(k) ) is called∑ along dimension k. The Hamming distance between two vertices x = x1 x2 . . . xn and y = y1 y2 . . . yn in i=n Qn is h(x, y) = i=1 |xi − yi |. Clearly, dQn (x, y) = h(x, y). Qn can be partitioned along some dimension i ∈ {1, 2, . . . , n} i,0 i,1 i,0 i,1 into two sub-cubes Qn−1 and Qn−1 , where Qn−1 and Qn−1 are induced by {x = x1 x2 . . . xi−1 1xi+1 . . . xn |x ∈ V (Qn )} and {x = x1 x2 . . . xi−1 0xi . . . xn |x ∈ V (Qn )}, respectively. Clearly, Qni,−11 and Qni,−01 are isomorphic to Qn−1 . If i is specified, then i ,1 i ,0 Qn−1 and Qn−1 are written as Qn0−1 and Qn1−1 , respectively for short. An n-dimensional folded hypercube is denoted by FQn = (V (FQn ), E(FQn )), where V (FQn ) = V (Qn ), E(FQn ) = E(Qn ) ∪ Ea , and Ea = {(u, u)|u = u1 u2 . . . un , u = u1 u2 . . . un ∈ V (FQn )}. Let Ei = {(u, u(i) )|u ∈ V (FQn )} for i ∈ {1, 2, . . . , n}. Readers may find the figures of FQ2 and FQ3 in Fig. 1. Definition 1 ([13]). FQn can be partitioned along dimension i ∈ {1, 2, . . . , n} into two (n − 1)-cubes Qn0−1 and Qn1−1 . Moreover, all edges in Ea are between Qn0−1 and Qn1−1 . In the following, there are some related lemmas. Lemma 1 ([33]). FQn − Ei ∼ = Qn for i ∈ {1, 2, . . . , n, a}. Let fe and fv be the number of faulty edges and faulty vertices in Qn . Please cite this article in press as: D. Cheng, Cycles embedding in folded hypercubes under the conditional fault model, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.02.020.

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Fig. 1. FQ2 and FQ3 , where dashed lines are in Ea .

Lemma 2 ([16,28]). In Qn with fv + fe ≤ n − 2 and n ≥ 3, every fault-free edge lies on a fault-free cycle of every even length ℓ with 4 ≤ ℓ ≤ 2n − 2fv . Lemma 3 ([35]). In Qn with fv + fe ≤ 2n − 5 and n ≥ 3 and under the conditional fault model, each edge of Qn − Fv − Fe lies on a fault-free cycle of every even length ℓ with 6 ≤ ℓ ≤ 2n − 2|Fv |. Lemma 4 ([24]). Let Fv and Fe be the set of faulty vertices and faulty edges in Qn with |Fv | + |Fe | ≤ n − 2 and n ≥ 3. Let u and v be two distinct fault-free vertices in Qn . There is a fault-free path of length ℓ between u and v in Qn , where dQn (u, v ) + 2 ≤ ℓ ≤ 2n − 2|Fv | − 1 and 2|(ℓ − dQn (u, v )). Lemma 5 ([28]). In Qn with fe ≤ n − 2, fe + fv ≤ 2n − 4 and n ≥ 5, Qn contains a fault-free cycle of every even length ℓ with 4 ≤ ℓ ≤ 2n − 2fv . Lemma 6 ([29]). Let (u, v ) and (x, y) be any two disjoint edges in Qn with n ≥ 2. Then there exist two disjoint paths P [u, v] and P [x, y] in Qn with odd lengths 1 ≤ ℓ(P [u, v]) ≤ 2n−1 − 1 and 1 ≤ ℓ(P [x, y]) ≤ 2n−1 − 1. Lemma 7 ([26]). Let u and v be any two vertices of Qn with dQn (u, v ) = d. Then there are n internally disjoint paths joining u and v in Qn , where d paths of them are of length d and lies in a d-dimensional subcube; the other paths are of length d + 2. Lemma 8 ([18]). In FQn with |FFv | + |FFe | ≤ n − 1 and n ≥ 3, FQn − FFv − FFe contains a fault-free cycle of every even length from 4 to 2n − 2|FFv |. In FQn with |FFv | + |FFe | ≤ n − 1 and n ≥ 4 is even, FQn − FFv − FFe contains a fault-free cycle of every odd length from n + 1 to 2n − 2|FFv | − 1. Lemma 9 ([20]). In FQn with |FFe | ≤ 2n − 3 and under the conditional fault model, if n ≥ 2, then FQn − FFe contains a cycle of every even length ℓ with 4 ≤ ℓ ≤ 2n ; if n ≥ 2 being even, FQn − FFe contains a cycle of every odd length ℓ′ with n + 1 ≤ ℓ′ ≤ 2n − 1. In an n-dimensional folded hypercube FQn , a vertex is k-free if it is incident to at most k fault-free edges. [14] Lemma 10 ([14]). In FQn with |FFv | + |FFe | ≤ 2n − 4 and n ≥ 3, there is at most one 2-free vertex. Note that Hsieh et al. [14] proved the following lemma in the proof of Theorem 1 in [14]. Lemma 11 ([14]). In FQn with |FFv | + |FFe | ≤ 2n − 4 and under the conditional faulty model, then FQn can be partitioned along dimension i ∈ {1, 2, . . . , n, a} of some faulty edge into Qn0−1 and Qn1−1 , such that each vertex in FQn − Ei is incident to at least two fault-free edges. By Lemmas 10 and 11, we know that there is at most one vertex in Qn0−1 (or Qn1−1 ) that is incident to only one fault-free edge of Qn0−1 (or Qn1−1 ). 3. Fault-free even cycles embedding in a faulty folded hypercube Let us start with another lemma. Lemma 12. In FQn with |FFe | + |FFv | ≤ 2n − 4 and |FFe | ≤ n − 1, FQn − FFv − FFe contains a fault-free cycle of every even length ℓ with 4 ≤ ℓ ≤ 2n − 2|FFv |, where n ≥ 5. Please cite this article in press as: D. Cheng, Cycles embedding in folded hypercubes under the conditional fault model, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.02.020.

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Proof. Case 1. |FFe | ≤ n − 2. Let H = FQn − Ea . By Lemma 1, H = FQn − Ea ∼ = Qn . Let Fv′ = FFv ∩ V (H), Fe′ = FFe ∩ E(H), Fa = FFe ∩ Ea . ′ ′ ′ Note that |Fe | ≤ |FFe | ≤ n − 2 and |Fv | + |Fe | ≤ |FFv | + |FFe | ≤ 2n − 4, by Lemma 5, H (⊂ FQn ) contains a fault-free cycle of every even length ℓ with 4 ≤ ℓ ≤ 2n − 2|Fv | (= 2n − 2|FFv |). Case 2. |FFe | = n − 1. There is some dimension i ∈ {1, 2, . . . , n, a} such that |Ei ∩ FFe | ≥ 1. By Lemma 1, FQn − Ei ∼ = Qn . Let H ′′ = FQn − Ei , Fi′′ = Ei ∩ FFe , Fe′′ = E(H ′′ ) ∩ FFe , and Fv′′ = V (H ′′ ) ∩ FFv . Then |Fi′′ | ≥ 1, |Fe′′ | = |FFe | − |Fi′′ | ≤ (n − 1) − 1 = n − 2 and |Fe′′ | + |Fv′′ | ≤ |FFe | + |FFv | ≤ 2n − 4. By Lemma 5, H ′′ (⊂ FQn ) contains a fault-free cycle of every even length ℓ with 4 ≤ ℓ ≤ 2n − 2|Fv′′ | (= 2n − 2|FFv |). Hence, the lemma holds. Lemma 13. In FQ4 with |FFe | + |FFv | ≤ 4 and under the conditional fault model, FQ4 − FFv − FFe contains a fault-free edge of every even length ℓ with 4 ≤ ℓ ≤ 16 − 2|FFv |. Proof. The proof of this lemma is in Appendix Our main result is the following theorem. Theorem 1. In FQn with |FFe | + |FFv | ≤ 2n − 4 and under the conditional fault model, FQn − FFv − FFe contains a fault-free cycle of every even length ℓ with 4 ≤ ℓ ≤ 2n − 2|FFv |, where n ≥ 3. Proof. By Lemma 8, the theorem holds for n = 3. By Lemma 13, the theorem is true for n = 4. By Lemma 12, the theorem is true for |FFe | + |FFv | ≤ 2n − 4 and |FFe | ≤ n − 1, where n ≥ 5. If |FFv | = 0, then |FFe | ≤ 2n − 4, by Lemma 9, the theorem is true. In the following, we need to prove the theorem holds for |FFe | + |FFv | ≤ 2n − 4, |FFe | ≥ n, and |FFv | ≥ 1, where n ≥ 5. Since |FFe | + |FFv | ≤ 2n − 4, |FFe | ≥ n and |FFv | ≥ 1, we have 1 ≤ |FFv | ≤ n − 4. If all the faulty edges are in Ea , then there is no faulty edge in FQn − Ea . By Lemma 1, FQn − Ea ∼ = Qn . By Lemma 2, there is a fault-free cycle of every even length from 4 to 2n − 2|FFv | in FQn − Ea ⊂ FQn . Assume that not all the faulty edges are in Ea . By Lemma 11, FQn can be partitioned along dimension i ∈ {1, 2, . . . , n, a} of some faulty edge into Qn1−1 and Qn0−1 such that each vertex in FQn − Ei is incident to at least two fault-free edges. Let j

j

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Fi = Ei ∩ FFe , Fa = Ea ∩ FFe , Fe = FFe ∩ Qn−1 and Fvj = FFv ∩ Qn−1 for j = 0, 1. So |Fi | ≥ 1, |Fe0 | + |Fv0 | ≤ 2n − 5 and |Fe1 | + |Fv1 | ≤ 2n − 5. Without loss of generality, we may assume that |Fe1 | + |Fv1 | ≤ |Fe0 | + |Fv0 |. Then |Fe1 | + |Fv1 | ≤

⌊ (2n−4)2 −|Fi | ⌋ ≤ ⌊ (2n−24)−1 ⌋ = n − 3. Since |Fe1 | + |Fv1 | ≤ n − 3 and the degree of Qn1−1 is n − 1, thus each vertex of Qn1−1 is incident to at least two fault-free edges. Note that there is at most one vertex in Qn0−1 incident to only one fault-free edge of Qn0−1 . We discuss the following cases. Case 1. |Fe0 | + |Fv0 | ≤ 2n − 7. Since |Fe1 | + |Fv1 | ≤ n − 3, by Lemma 2 there is a fault-free cycle of every even length ℓ with 4 ≤ ℓ ≤ 2n−1 − 2|Fv1 | in Qn1−1 . In the following, we will find the desired cycles of every even length ℓ with 2n−1 − 2|Fv1 | + 2 ≤ ℓ ≤ 2n − 2|FFv | in FQn . ℓ By Lemma 2, there is a fault-free cycle C1 of even length ℓ1 = 2n−1 − 2|Fv1 | in Qn1−1 . Since ⌊ 21 ⌋ − |Fi | − |Fa | − |Fv0 | − |Fe0 | = 2n−1 −2|Fv1 | 2 (5−2)

⌋ − |Fi | − |Fa | − |Fv0 | − |Fe0 | = 2n−2 − |Fv1 | − |Fi | − |Fa | − |Fv0 | − |Fe0 | ≥ 2n−2 − (|FFv | + |FFe |) ≥ 2n−2 − (2n − 4) ≥ 2 − (2 × 5 − 4) = 2 for n ≥ 5. We can find two disjoint fault-free edges, denoted by (a, b) and (c , d), in C1 , such that the edges in {(a, a(i) ), (b, b(i) ), (a(i) , b(i) ), (c , c (i) ), (d, d(i) ), (c (i) , d(i) )} or {(a, a), (b, b), (a, b), (c , c), (d, d), (c , d)} are ⌊

fault-free. (Without loss of generality, we assume that the former is true.) (If Qn0−1 contains a vertex that is incident to only one fault-free edge in Qn0−1 , in the above discussion, we need to denote this vertex by a(i) , and find one faulty edge, denoted by (a(i) , b(i) ), such that (a, a(i) ), (b, b(i) ) are fault-free. Because a(i) is incident to n − 2 faulty edges in Qn0−1 and there are at most n − 3 other faulty elements not incident to a(i) , we can find such faulty edge. In the following discussion of this paragraph, we take (a(i) , b(i) ) as a fault-free edge temporarily.) Since |Fe0 | + |Fv0 | ≤ 2n − 7, by Lemma 3, (a(i) , b(i) ) lies on a fault-free cycle C0 of even length ℓ0 in Qn0−1 , where 6 ≤ ℓ0 ≤ 2n−1 − 2|Fv0 |. Let C = C1 − (a, b) + (a, ai ) + (b, b(i) ) + C0 − (a(i) , b(i) ) with length ℓ = ℓ1 − 1 + 2 + ℓ0 − 1 = ℓ1 + ℓ0 . Hence, 2n−1 − 2|Fv1 | + 6 ≤ ℓ ≤ 2n − 2|FFv |. Let C ′ = C1 − (a, b) + (a, a(i) ) + (b, b(i) ) + (a(i) , b(i) ) with length ℓ′ . Then ℓ′ = ℓ1 − 1 + 3 = ℓ1 + 2 = 2n−1 − 2|Fv1 | + 2. Let C ′′ = C1 − (a, b) − (c , d) + (a, a(i) ) + (b, b(i) ) + (a(i) , b(i) ) + (c , c (i) ) + (d, d(i) ) + (c (i) , d(i) ) with length ℓ′′ . Then ′′ ℓ = ℓ1 − 2 + 6 = ℓ1 + 4 = 2n−1 − 2|Fv1 | + 4. Case 2. |Fe0 | + |Fv0 | = 2n − 6. Since |Fi | ≥ 1, |Fe1 | + |Fv1 | ≤ 1 < (n − 1) − 2 for n ≥ 5, by Lemma 2 there is a fault-free cycle of every even length from 4 to 2n−1 − 2|Fv1 | in Qn1−1 ⊂ FQn . Note that |Fe0 | + |Fv0 | = 2n − 6 ≥ 2 × 5 − 6 = 4 for n ≥ 5 and |Fi | + |Fa | + |Fv1 | + |Fv1 | ≤ 2 in this case, there is a faulty edge, denoted by (x, y), in Qn0−1 , such that (x, x(i) ), (y, y(i) ) and (x(i) , y(i) ) are fault-free edges. (Note that if there is a vertex in Qn0−1 that is incident to only one fault-free edge in Qn0−1 , we can denote this vertex by x, where (x, y) is a faulty edge that satisfies the above conditions.) Taking (x, y) as a fault-free edge in Qn0−1 temporarily, by Lemma 3, Please cite this article in press as: D. Cheng, Cycles embedding in folded hypercubes under the conditional fault model, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.02.020.

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(x, y) lies on a fault-free cycle C0 of even length ℓ0 in Qn0−1 − (Fv0 + Fe0 ) + {(x, y)}, where 6 ≤ ℓ0 ≤ 2n−1 − 2|Fv0 |. Since |Fv1 | + |Fe1 | ≤ 1, by Lemma 2, (x(i) , y(i) ) lies on a fault-free cycle C1 of even length ℓ1 in Qn1−1 , where 4 ≤ ℓ1 ≤ 2n−1 − 2|Fv1 |. Let C = C0 − (x, y) + (x, x(i) ) + (y, y(i) ) + C1 − (x(i) , y(i) ) with length ℓ = ℓ0 − 1 + 2 + ℓ1 − 1 = ℓ0 + ℓ1 . Hence, 10 ≤ ℓ ≤ 2n − 2|FFv |. Note that 2n−1 − 2|Fv1 | + 2 ≥ 2n−1 − 2(|Fe1 | + |Fv1 |) + 2 ≥ 25−1 − 2 × 1 + 2 ≥ 24 = 16 > 10 for n ≥ 5. Hence there is a fault-free cycle C ′ of even length ℓ′ in FQn , where 10 < 2n−1 − 2|Fv1 | + 2 ≤ ℓ′ ≤ 2n − 2|FFv |. Case 3. |Fe0 | + |Fv0 | = 2n − 5. Since |Fi | ≥ 1 and |FFe | + |FFv | ≤ 2n − 4, in this case |Fe1 | + |Fv1 | = 0 and |Fi | = 1. By Lemma 2, there is a fault-free cycle C1 of even length ℓ1 in Qn1−1 ⊂ FQn , where 4 ≤ ℓ1 ≤ 2n−1 . Note that |Fe1 | = 0, |Fi | = 1 and |FFe | ≥ n, so |Fe0 | ≥ n − 1 ≥ 4 for n ≥ 5. We can find two disjoint faulty edges in Qn0−1 , denoted by (a, b) and (c , d), such that the edges in {(a, a), (b, b), (c , c), (d, d)} or {(a, a(i) ), (b, b(i) ), (c , c (i) ), (d, d(i) )} are fault-free. Without loss of generality, we may assume that the former is true. (Note that if Qn0−1 contains a vertex that is incident to only one fault-free edge in Qn0−1 , we can denote this vertex by a, where (a, b) is a faulty edge that satisfies the above conditions.) Taking (a, b) and (c , d) as fault-free edges temporarily, by Lemma 3, there is a cycle C0 of even length ℓ0 in Qn0−1 − (Fe0 + Fv0 ) + {(a, b), (c , d)}, where ℓ0 = 2n−1 − 2|Fv0 |. (Assume that C0 contains (a, b) and (c , d), otherwise we can take two fault-free disjoint edges in C0 instead of (a, b) and (c , d).) Note that |Fv1 | = |Fe1 | = 0, by Lemma 6, there exist two disjoint fault-free paths P1 of length ℓ′1 joining a and b and P2 of length ℓ′′1 joining c and d, where 1 ≤ ℓ′1 ≤ 2n−2 − 1 and 1 ≤ ℓ′′1 ≤ 2n−2 − 1. Let C = C0 − (a, b) − (c , d) + (a, a) + (b, b) + (c , c) + (d, d) + P1 + P2 with length ℓ = ℓ0 − 2 + 4 + ℓ′1 + ℓ′′1 = ℓ0 + 2 + ℓ′1 + ℓ′′1 . Hence 2n−1 − 2|Fv0 | + 4 ≤ ℓ ≤ 2n − 2|Fv0 |. Note that |Fv0 | = |FFv | ≥ 1, so 2n−1 − 2|Fv0 | + 4 = 2n−1 − 2|FFv | + 4 ≤ 2n−1 − 2 + 4 = 2n−1 + 2. Hence, there is a fault-free cycle C ′ of even length ℓ′ in FQn , where 2n−1 + 2 ≤ ℓ′ ≤ 2n − 2|FFv |, and C ′ is a desired cycle. By the above cases, the theorem holds. 4. Fault-free odd cycles embedding in a faulty folded hypercube Theorem 2. Let FFv and FFe be the set of faulty vertices and faulty edges in FQn with |FFv | + |FFe | ≤ 2n − 5 and under the conditional fault model. There is a fault-free odd cycle of length ℓ with n + 1 ≤ ℓ ≤ 2n − 2|FFv | − 1 in FQn , where n ≥ 4 is even. Proof. By Lemma 8, the theorem is true for n = 4. We consider n ≥ 6 as follows. Case 1. |FFe | ≥ 1. By Lemma 11, FQn can be partitioned along some dimension i ∈ {1, 2, . . . , n, a} of some faulty edge into Qn0−1 and 1 Qn−1 such that each vertex in FQn − Ei is incident to at least two fault-free edges. Let Fi = FFe ∩ Ei , Fa = Ea ∩ FFe , Fei = FFe ∩ Qni −1 and Fvi = FFv ∩ FQn for i = 0, 1. Without loss of generality, we may assume that |Fe0 | + |Fv0 | ≤ |Fe1 | + |Fv1 |. Then |Fe0 | + |Fv0 | ≤ n − 3 and |Fe1 | + |Fv1 | ≤ 2n − 6. Since the degree of Qn0−1 is n − 1, each vertex in Qn0−1 is incident to at least two fault-free edges of Qn0−1 . Note that there is at most one vertex in Qn1−1 incident to only one fault-free edge of Qn1−1 . Case 1.1. |Fe1 | + |Fv1 | ≤ 2n − 7. Since there are 2n−2 vertex-disjoint edges in Qn1−1 and 2n−2 > 2n − 5 ≥ |FFe | + |FFv | for n ≥ 6, there is a fault-free edge, say e = (a, b), in Qn1−1 such that (a, a), (b, b), (a, a(i) ) and (b, b(i) ) are fault-free. Note that dQ 0 (a, b(i) ) = n − 2. By Lemma 7, n there are n − 2 internally disjoint paths of length n − 2 between a and b(i) . Since |Fv0 | + |Fe0 | ≤ n − 3, there is at least one fault-free path P0′ of length n − 2 between a and b(i) . Hence P0′ + (a, a) + (b(i) , b) + e is a fault-free cycle of length n + 1. By Lemma 4, there is a fault-free path P0 of even length ℓ0 between a and b(i) in Qn0−1 , where n ≤ ℓ0 ≤ 2n−1 − 2|Fv0 | − 2. Then P0 + (a, a) + (b, b(i) ) + e is a fault-free cycle of odd length from n + 3 to 2n−1 − 2|Fv0 | + 1. Case 1.1.1. Each vertex in Qn1−1 is incident to at least two fault-free edges. By Lemma 3, there is a fault-free cycle C1 of even length ℓ1 with 6 ≤ ℓ1 ≤ 2n−1 − 2|Fv1 | containing e in Qn1−1 . Then C1 − e + (a, a) + (b, b(i) ) + P0 is a fault-free cycle of odd length from n + 7 to 2n − 2|FFv | − 1. Since 2n−1 − 2|Fv0 | + 1 ≥ 2n−1 − 2(n − 3) + 1 > n + 7 for n ≥ 6, we can find a fault-free cycle of odd length from n + 3 to 2n − 2|FFv | − 1. Case 1.1.2. There is a vertex, say x ∈ V (Qn1−1 ), is incident to only one fault-free edge, say (y, x), of Qn1−1 . Then (x, x) or (x, x(i) ) is fault-free. We assume that the former is true. Since x is incident to n − 2 faulty edges of Qn1−1 , there is at least one faulty edge, say (x, z), such that (z , z (i) ) is fault-free. Take (x, z) as a fault-free edge temporarily. By Lemma 3, there is a cycle C1′ of even length ℓ′1 with 6 ≤ ℓ′1 ≤ 2n−1 − 2|Fv1 | containing (x, z). Obviously, C1′ contains (y, x) and C1′ − (x, z) is fault-free. Let ⟨y, x, z ⟩ be a 2-path in C1′ . Note that dQ 0 (x, z (i) ) = n − 2. By Lemma 4, there is a fault-free path P0′′ of even n−1

length ℓ′′0 between z (i) and x, where n − 2 ≤ ℓ′′0 ≤ 2n−1 − 2|Fv0 | − 2. Then C1′ − (z , x) + (z , z (i) ) + (x, x) + P0′′ is a fault-free cycle of odd length from n + 5 to 2n − 2|FFv | − 1. Case 1.2. |Fe1 | + |Fv1 | = 2n − 6. The proof of this case is divided into two cases according to the odd length ℓ of a desired fault-free cycle. Case 1.2.1 n + 1 ≤ ℓ ≤ 2n−1 − 2|Fv0 | + 1. The proof of this case is similar to Case 1.1. Case 1.2.2. 2n−1 − 2|Fv0 | + 3 ≤ ℓ ≤ 2n − 2|FFv | − 1. Take a faulty edge, say e′ = (c , d), in Qn1−1 as a fault-free edge temporarily. Note that |Fi | + |Fa | = 1 and |FFv0 | + |FFe0 | = 0, the edges of {(c , c), (d, d(i) )} or {(c , c (i) ), (d, d)} are fault-free. Without loss of generality, we may assume that the former is Please cite this article in press as: D. Cheng, Cycles embedding in folded hypercubes under the conditional fault model, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.02.020.

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true. If each vertex is incident to at least two fault-free edges, then taking e′ instead of e in Case 1.1.1. If there is a vertex that is incident to only one fault-free edge, then taking c as this vertex and choosing e′ = (c , d) being a faulty edge such that (d, d(i) ) is fault-free. Take e′ instead of (x, z) in Case 1.1.2. The rest proof of this case is similar to Case 1.1.2. Case 2. |FFe | = 0, i.e., |FFv | ≤ 2n − 5. By Lemma 11, FQn can be partitioned along some dimension i ∈ {1, 2, . . . , n, a} of some fault-free edge, say e′′ , into Qn0−1 and Qn1−1 such that each vertex in FQn − Ei is incident to at least two fault-free edges and Qn0−1 contains at least one faulty vertex. (Note that |{e′′ }| + |FFv | ≤ 1 + (2n − 5) = 2n − 4, if taking e′′ as a faulty edge temporarily, we can find this edge.) Let Fvi = FFv ∩ Qni −1 for i = 0, 1. Assume that |Fv0 | ≤ |Fv1 |, then 1 ≤ |Fv0 | ≤ n − 3 and |Fv1 | ≤ 2n − 6. We further consider the following cases. Case 2.1. n + 1 ≤ ℓ ≤ n + 3. The proof of this case is similar to Case 1.1. Case 2.2. 2n−1 − 2|Fv0 | + 1 ≤ ℓ ≤ 2n − 2|FFv | − 1. Since |Fe1 | = 0 < n − 2 and |Fe1 | + |Fv1 | ≤ 2n − 6 = 2(n − 1) − 4 and n − 1 ≥ 5 for n ≥ 6, by Lemma 5, there is a fault-free ℓ′′

cycle C1′′ of even length ℓ′′1 with 4 ≤ ℓ′′1 ≤ 2n−1 − 2|Fv1 | in Qn1−1 . Since there are ⌊ 21 ⌋ ≥ 2 disjoint edges in C1′′ and |Fv0 | = 1, we can find at least one edge, say (x, y), in C1′′ such that (x, x(i) ) and (y, y) are fault-free. Note that dQ 0 (x(i) , y) = n − 2, n −1

by Lemma 4, there is a fault-free path P0′′ of even length ℓ′′0 between x(i) and y, where n ≤ ℓ′′0 ≤ 2n−1 − 2|Fv0 | − 2. Hence P0′′ + C1′′ − (x, y) + (x, x(i) ) + (y, y) is a fault-free cycle with odd length from n + 5 to 2n − 2|FFv | − 1. By the above cases, the theorem is true. 5. Concluding remarks In this paper, we consider an n-dimensional folded hypercube FQn under the conditional fault model, and prove that if

|FFe | + |FFv | ≤ 2n − 4 and n ≥ 3, then there exists a fault-free cycle of every even length from 4 to 2n − 2|FFv | in FQn ; if |FFe | + |FFv | ≤ 2n − 5 and n ≥ 4 is even, there exists a fault-free cycle of every odd length ℓ from n + 1 to 2n − 2|FFv | − 1. Note that if there are |FFv | = 2 × 3 − 3 = 3 vertices in FQ3 , then 4 > 23 − 2 × 3 = 2, which is a contradiction to the even length from 4 to 2n − 2|FFv |. Hence, the bound 2n − 4 is best. Acknowledgments This work is supported by the Tian Yuan Fund of National Natural Science Foundation of China (Grant No. 11626114), the Natural Science Foundation of Guangdong Province, China (Grant No. 2016A030310082) and the Fundamental Research Funds for the Central Universities (Grant No. 21616311). Appendix The proof of Lemma 13. Proof. By Lemma 8, the lemma holds for |FFv | + |FFe | ≤ 3. We consider |FFv | + |FFe | = 4 as follows. Case 1. |FFe | ≥ 1. By Lemma 11, FQ4 can be partitioned along dimension i ∈ {1, 2, 3, 4, a} some faulty edge into Q31 and Q32 , such that each vertex in FQ4 − Ei is incident to at least two fault-free edges. Note that FQ4 − Ei ∼ = Q4 and the number of faulty edges and faulty vertices is no more than 3, by Lemma 3, there is a fault-free cycle of every even length from 6 to 16 − 2|FFv |. Note that, there are four vertex-disjoint 4-cycles in FQ4 − Ei , and there are at most three faulty vertices and edges in FQ4 − Ei , so we can always find a fault-free 4-cycle in FQ4 − Ei . Case 2. |FFe = 0|, i.e., |FFv | = 4. Let C41 = ⟨0000, 0001, 0011, 0010, 0000⟩, C42 = ⟨0100, 0101, 0111, 0110, 0100⟩, C43 = ⟨1000, 1001, 1011, 1010, 1000⟩, 4 C4 = ⟨1100, 1101, 1111, 1110, 1100⟩ and I4 = {C41 , C42 , C43 , C44 }. Note that C41 , C42 , C43 and C43 are vertex-disjoint 4-cycles. If these four faulty vertices are all in one 4-cycle of I4 , then two 4-cycles of I4 induce a Q3i , where i ∈ {0, 1}. Obviously, there is a fault-free cycle of even length ℓ with ℓ = 4, 6, 8 in Q3i ⊂ FQ4 . If these four faulty vertices are in two 4-cycles of I4 , then we also can find cycles of lengths 4, 6, 8 in FQ4 . Hence we need to consider the following two subcases. Case 2.1. These four faulty vertices are distributed in three 4-cycles of I4 . In this case, two 4-cycles of I4 contain only one faulty vertex respectively, and the other 4-cycle contains two faulty vertices. FQ4 is a central symmetric graph. From Figs. 2 to 4, we can find that there are isomorphic graphs. Hence, we only need to consider the case that C41 and C42 contain only one faulty vertex respectively, and C43 contains two faulty vertices. Note that C44 is fault-free. Let the faulty vertices of C41 and C42 be u and v , respectively. Since FQ4 is vertex-transitive, we may assume that u = 0000. We can find cycles of lengths 4, 6, 8 in the graph induced by C41 , C42 and C44 . (See Table 1.) Note that if u = 0000 and v = 0101, the desired cycles of even lengths 4,6,8 are the same as the case that u = 0000 and v = 0100. Case 2.2. These four faulty vertices are distributed in four 4-cycles of I4 . Please cite this article in press as: D. Cheng, Cycles embedding in folded hypercubes under the conditional fault model, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.02.020.

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Table 1 Fault-free cycles of lengths 4, 6, 8 in FQ4 . u

v

Cycles of lengths 4, 6, 8

0000

0100

0000

0111

0000

0110

⟨1001, 1011, 1010, 1000, 1001⟩ ⟨1011, 0011, 0010, 1010, 1000, 1001, 1011⟩ ⟨1011, 0011, 0111, 0110, 0010, 1010, 1000, 1001, 1011⟩ ⟨1011, 1010, 1000, 1001, 1011⟩ ⟨1011, 0011, 0010, 1010, 1000, 1001, 1011⟩ ⟨1011, 0011, 0001, 0101, 0100, 0110, 0010, 1010, 1011⟩ ⟨1011, 1010, 1000, 1001, 1011⟩ ⟨1011, 0011, 0010, 1010, 1000, 1001, 1011⟩ ⟨1011, 1001, 0001, 0101, 0111, 0011, 0010, 1010, 1011⟩

Table 2 Fault-free cycles of lengths 4, 6, 8 in FQ4 . x

y

Cycles of lengths 4, 6, 8

1001

1110

1011

1100

1010

1101

1000

1111

⟨1011, 0011, 0010, 1010, 1011⟩ ⟨0001, 0101, 0100, 0110, 0010, 0011, 0001⟩ ⟨1011, 0011, 0001, 0101, 0100, 0110, 0010, 1010, 1011⟩ ⟨1010, 1110, 0110, 0010, 1010⟩ ⟨1001, 0001, 0011, 0010, 1010, 1000, 1001⟩ ⟨1001, 1101, 1111, 1110, 0001, 0101, 0100, 0110, 1001⟩ ⟨1100, 1110, 0001, 0011, 1100⟩ ⟨1001, 0110, 0010, 0011, 1100, 1000, 1001⟩ ⟨1100, 1110, 0001, 0101, 0100, 0110, 1001, 1000, 1100⟩ ⟨0001, 0011, 1011, 1001, 0001⟩ ⟨1110, 1010, 0010, 0110, 0100, 1100, 1110⟩ ⟨0001, 0011, 0010, 0110, 1110, 1100, 1101, 1001, 0001⟩

Fig. 2. Fig. 2(a) is isomorphic to Fig. 2(b).

Without loss of generality, we may assume that the four vertices in C41 , C42 , C43 and C44 are u, v, x, y, respectively. Since FQ4 is vertex-transitive, we may assume that u = 0000. Note that if v = 0100 or v = 0101 then u and v are in a 4-cycle, by the isomorphism of FQ4 (see Fig. 2), this case is similar to Case 2.1. If v = 0110, then u and v are in a 4-cycle, by the isomorphism of FQ4 (see Fig. 3), this case is similar to Case 2.1. Hence v = 0111. With the similar discussions, x and y cannot be in the Please cite this article in press as: D. Cheng, Cycles embedding in folded hypercubes under the conditional fault model, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.02.020.

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Fig. 3. Fig. 3(a) is isomorphic to Fig. 3(b).

Fig. 4. Fig. 4(a) is isomorphic to Fig. 4(b).

same 4-cycle of Q31 . Hence, x and y are in {1001, 1110}, or {1011, 1100}, or {1010, 1101}, or {1000, 1111}. We can find cycles of lengths 4, 6, 8 in FQ4 . (See Table 2.) By the above cases, the lemma holds. Please cite this article in press as: D. Cheng, Cycles embedding in folded hypercubes under the conditional fault model, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.02.020.

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Please cite this article in press as: D. Cheng, Cycles embedding in folded hypercubes under the conditional fault model, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.02.020.