Fault-Tolerant Vertex-Pancyclicity of Crossed Cubes CQn Xirong Xu School of Computer Science and Technology Dalian University of Technology, P.R.China 2018 - 5 - 21

OutLine 一 二 三 四 五 Fault-Tolerant Vertex-Pancyclicity of CQn Definitions & Properties Abstract Introduction Remarks 一 二 三 四 五

Abstract 一 1 Abstract The pancyclicitiy is an important property to determine if topology of a network is suitable for some applications where mapping cycles or paths of any length into the topology of the network is required. The n-dimensional crossed cubes CQn is an important variant of the hypercube Qn, which possesses some properties suprior to the hypercube. In this talk, we investigate the fault-tolerant vertex-pancyclicity of CQn and shows that if CQn(n≥3) contains at most n-3 faulty vertices and/or edges then, for any fault-free vertex u and any integer l with 6≤l≤ 2n - fv except l=7, there is a fault-free cycle of length l containing the vertex u, where fv is the number of faulty vertices. The result is optimal in some senses. of 1 27

2 Introduction 二 Introduction A graph G of order n is k-pancyclic(k≤n) if it contains cycles of every length from k to n inclusive, and G is pancyclic if it is g-pancyclic, where g=g(G) is the girth of G. A graph is of pancyclicity if it is pancyclic. The pancyclicity, which means the hamiltonicity, is an important property to determine if a topology of a network is suitable for some applications where mapping cycles of any length into the topology of the network is required. The concept of pancyclicity has been extended to vertex-pancyclicity and edge-pancyclicity. A graph G of order n is vertex-pancyclic(resp. edge-pancyclic) if any vertex(resp.edge) lies on cycles of every length from g(G) to n inclusive. Fig.1 Containment relationships of Hamiltonian-like properties. A:Hamiltonian; B:Hamiltonian-connected; C:pancyclic; D:vertex-pancyclic; E:edge-pancyclic; F:panconnected of 2 27

2 Introduction 二 Introduction A large network system in the daily operation, it will inevitably be a variety of errors, so for a good network, not only in all normal circumstances to ensure the normal operation of the system, but also in the network after a certain error, the system can still ensure the remaining part of the normal operation. Therefore, in the measurement of a network as well as to consider the network topology graph of fault tolerance. A graph G is k-fault-tolerant pancyclic if G-F remains pancyclic for any F ⊂ V (G)∪E(G) with |F|≤ k; A graph G is k-vertex-fault-tolerant pancyclic if G-F remains pancyclic for any F ⊂ V (G) with |F|≤ k; A graph G is k-edge-fault-tolerant pancyclic if G-F remains pancyclic for any F ⊂ E(G) with |F|≤ k. In this talk, we investigate the fault-tolerant vertex-pancyclicity of CQn, and shows that if CQn(n≥3) contains at most n-3 faulty vertices and/or edges then, for any fault-free vertex u and any integer l with 6≤l≤ 2n - fv except l=7, there is a fault-free cycle of length l containing the vertex u, where fv is the number of faulty vertices. of 3 27

3 Definitions and Properties Definitions&Properties 三 3 Definitions and Properties Definition 3.1. Two binary string x=x2x1 and y=y2y1 are pair-related, denoted by x~y, if an only if (x, y)∈ {(00,00),(10,10),(01,11),(11,01)}. Definition 3.2. An n-dimensional crossed cube , represented by CQn, is such an undirected graph, its vertex set remains similar such as the vertex set of Qn, two vertices first one is x=xnxn-1…x2x1 and other y=ynyn-1…y2y1 are connected by means of an undirected edge if and most effective if there exists some j(1≤j≤n) such that Definition 3.3. The vertex is u=0un-2un-1…u1u0 ∈ V(CQ0n-1) and vertex is v=0vn-2vn-1…v1v0 ∈ V(CQ1n-1) are adjacent in CQn if and only if of 4 27

3 Definitions and Properties Definitions&Properties 三 3 Definitions and Properties Fig.2. CQ3 and CQ4 According to the above definition, it is not difficult to see that CQn is a n-regular graph with 2n vertices and n2n-1 deges. From the definition, CQn can be expressed as the union of two disjoint copies of CQn-1 by adding a perfect matching between them according to the specified rule. In other words, where and of 5 27

三 Definitions&Properties For two bits of the binary string xy, we use to denote its pair-related string. We can see that there are a lot of complicated relationships between two bits of the string labeled the vertex as the following table. of 6 27

三 Definitions&Properties Fig.3. 4-cycle,5-cycle and 6-cycle       Fig.3. 4-cycle,5-cycle and 6-cycle through u=00001 in CQ5 of 7 27

Definitions&Properties 三       of 8 27

三 Definitions&Properties Proof. Case1. n is odd. Subcase1.2 j=2k+1. of   Proof. Case1. n is odd. Subcase1.2 j=2k+1.       of 9 27

三 Definitions&Properties Proof. Case2. n is even. Subcase2.1 j=2k.   Proof. Case2. n is even. Subcase2.1 j=2k. By a similar argument in Case1.2, we can prove that the distance between uR and vR is 2 or 3. Subcase2.2 j=2k+1. By a similar argument in Case1.1, we can prove that the distance between uR and vR is 1 or 2 or 3. Fig.4. 4-cycle,5-cycle and 6-cycle through u=000001 in CQ6 of 10 27

三 Definitions&Properties J.Fan, Xu and Ma, Yang and Megson gave the following result, respectively: Lemma3.2 For any two vertices x and y with distance d in CQn with n≥2, CQn contains an xy-path of every length l from d to 2n-1 except for d+1. J.Fan and X.Lin proved the following result: Lemma3.3 The n-dimensional crossed cube CQn is vertex-pancyclic. Ma and Xu proved the following result: Lemma3.4 In CQn with fv faulty vertices and fe faulty edges, there exists a fault-free path of length l between any two distinct fault-free vertices for each l satisfying 2n-1-1 ≤ l ≤ 2n – fv - 1, provided that fv + fe ≤ n-3. of 11 27

三 Definitions&Properties Lemma3.5 If |F|=n-3 and F  L ,then for any vertex x in L，there are (n-2) 6-cycles that is disjoint in L containing x in CQn for n≥4. Proof. For any vertex x=xn-1xn-2…x2x1x0 in CQn for n ≥4, we divide the proof into two cases according to the parity of n. Case1. n is odd. (1) For each i=2k, 0≤k ≤ (n-4)/2, we can construct (n-2) 6-cycles that is disjoint in L(or R) containing vertex x as follows. of 12 27

三 Definitions&Properties Lemma3.5 If |F|=n-3 and F  L ,then for any vertex x in L，there are (n-2) 6-cycles that is disjoint in L containing x in CQn for n≥4. Proof. For any vertex x=xn-1xn-2…x2x1x0 in CQn for n ≥4, we divide the proof into two cases according to the parity of n. Case1. n is odd. (2) For each i=2k+1, 0≤k ≤ (n-5)/2, we can construct (n-2) 6-cycles that is disjoint in L(or R) containing vertex x as follows. of 13 27

三 Definitions&Properties Lemma3.5 If |F|=n-3 and F  L ,then for any vertex x in L，there are (n-2) 6-cycles that is disjoint in L containing x in CQn for n≥4. Proof. For any vertex x=xn-1xn-2…x2x1x0 in CQn for n ≥4, we divide the proof into two cases according to the parity of n. Case1. n is odd. (3) For i=n-3, we can construct (n-2) 6-cycles that is disjoint in L(or R) containing vertex x as follows. of 14 27

三 Definitions&Properties Lemma3.5 If |F|=n-3 and F  L ,then for any vertex x in L，there are (n-2) 6-cycles that is disjoint in L containing x in CQn for n≥4. Proof. Case1. n is odd. For example, for vertex x=00000 in CQ5 , there exist three 6-cycles that are disjoint in L containing vertex x=00000 as follows. Fig.5. Three 6-cycles are disjoint in L containing x=00000 in CQ5 of 15 27

三 Definitions&Properties Lemma3.5 If |F|=n-3 and F  L ,then for any vertex x in L，there are (n-2) 6-cycles that is disjoint in L containing x in CQn for n≥4. Proof. For any vertex x=xn-1xn-2…x2x1x0 in CQn for n ≥4, we divide the proof into two cases according to the parity of n. Case2. n is even. (1) For each i=2k, 0≤k ≤ (n-3)/2, we can construct (n-2) 6-cycles that is disjoint in L(or R) containing vertex x as follows. of 16 27

三 Definitions&Properties Lemma3.5 If |F|=n-3 and F  L ,then for any vertex x in L，there are (n-2) 6-cycles that is disjoint in L containing x in CQn for n≥4. Proof. For any vertex x=xn-1xn-2…x2x1x0 in CQn for n ≥4, we divide the proof into two cases according to the parity of n. Case2. n is even. (2) For each i=2k+1, 0≤k ≤ (n-4)/2, we can construct (n-2) 6-cycles that is disjoint in L(or R) containing vertex x as follows. of 17 27

三 Definitions&Properties Lemma3.5 If |F|=n-3 and F  L ,then for any vertex x in L，there are (n-2) 6-cycles that is disjoint in L containing x in CQn for n≥4. Proof. For any vertex x=xn-1xn-2…x2x1x0 in CQn for n ≥4, we divide the proof into two cases according to the parity of n. Case2. n is even. For example, for vertex x=000000, there exists four 6-cycles that are disjoint in L containing vertex x=000000 as follows. Fig.6. Four 6-cycles are disjoint in L containing x=000000 in CQ6 of 18 27

4 Fault-Tolerant Vertex-Pancyclicity of CQn 四 4 Fault-Tolerant Vertex-Pancyclicity of CQn Theorem1. If fv+fe≤n-3 and n≥3 then, for any fault-free vertex u in CQn and any integer l with 6 ≤ l ≤ 2n-fv except l=7, there is a fault-free cycle of length l containing the vertex u in CQn.   of 19 27

4 Fault-Tolerant Vertex-Pancyclicity of CQn 四 Fault-Tolerant Vertex-Pancyclicity of CQn 4 Fault-Tolerant Vertex-Pancyclicity of CQn Theorem1. If fv+fe≤n-3 and n≥3 then, for any fault-free vertex u in CQn and any integer l with 6 ≤ l ≤ 2n-fv except l=7, there is a fault-free cycle of length l containing the vertex u in CQn. Proof. Case1. |FL|=n-3. In this case, |FC|=|FR|=0. Since CQn is n-regular and |F|=n-3, there exists a fault-free edge joined with u, denoted by uu1. Let l=l’+1, then 2n-1-1 ≤ l’ ≤ 2n-fv-1. By Lemma3.4, there exists a fault-free Path P(u,u1) of length l’ in CQn. Thus, we only need to consider l with 6 ≤ l ≤ 2n-1-1 except l=7. We only need to consider that fault-free vertex u is in L. Otherwise if the fault-free vertex u is in R, then |FR|=0 and by Lemma3.3, There is a fault-free cycle of length l containing the vertex u for any l with 6 ≤ l ≤ 2n-1 except l=7. of 20 27

4 Fault-Tolerant Vertex-Pancyclicity of CQn 四 Fault-Tolerant Vertex-Pancyclicity of CQn 4 Fault-Tolerant Vertex-Pancyclicity of CQn Theorem1. If fv+fe≤n-3 and n≥3 then, for any fault-free vertex u in CQn and any integer l with 6 ≤ l ≤ 2n-fv except l=7, there is a fault-free cycle of length l containing the vertex u in CQn.   of 21 27

4 Fault-Tolerant Vertex-Pancyclicity of CQn 四 Fault-Tolerant Vertex-Pancyclicity of CQn 4 Fault-Tolerant Vertex-Pancyclicity of CQn Theorem1. If fv+fe≤n-3 and n≥3 then, for any fault-free vertex u in CQn and any integer l with 6 ≤ l ≤ 2n-fv except l=7, there is a fault-free cycle of length l containing the vertex u in CQn.   of 22 27

4 Fault-Tolerant Vertex-Pancyclicity of CQn 四 Fault-Tolerant Vertex-Pancyclicity of CQn 4 Fault-Tolerant Vertex-Pancyclicity of CQn Theorem1. If fv+fe≤n-3 and n≥3 then, for any fault-free vertex u in CQn and any integer l with 6 ≤ l ≤ 2n-fv except l=7, there is a fault-free cycle of length l containing the vertex u in CQn.   of 23 27

5 Some Remarks 五 Some Remarks We make some remarks on the optimality of our result in the following sense. (1) For example, in CQ5, taking u=00000, the cycles of length 4 containing the vertex u are as Table2 (we calculate it by computer program): If the vertex x=00010 and the vertex y=01000 are faulty, then there are no fault-free cycles of length 4 containing the vertex u in CQ5. of 24 27

5 Some Remarks 五 Some Remarks (2) For example, in CQ5, taking u=00001, the cycles of length 5 containing the vertex u are as Table3 (we calculate it by computer program): If the vertex x=00000 and the edge e=(00010,00011) is faulty, then there are no fault-free cycles of length 5 containing the vertex u in CQ5. of 25 27

5 Some Remarks 五 Some Remarks (3) As for the condition fv+fe≤n-3, we can say that it can be not improved as n-2 at least when n is small. In fact, if so, in CQ3, let u=000 be a faulty vertex, then there are no fault-free cycles of length 6 containing vertex 010. Our proof for Theorem1 uses induction on n≥3. The induction is based upon n=3, which does not hold for fv+fe=n-2 by the above example. The induction steps strongly depend on Lemma3.4 which holds only when fv+fe≤n-3. Thus, our method can not improve n-3 as n-2. However, as our further work, we must make it clear whether or not n-3 can be improved as n-2 for more general integer n. of 26 27

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