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

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

3. 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

4. 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

5. 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

6. 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 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

7. 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

8. 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

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

10. Definitions&Properties三 of 8 27

11. 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

12. 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 cycle,5-cycle and 6-cycle
through u= in CQ6 of 10 27

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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= as follows.
Fig.6. Four 6-cycles are disjoint in L
containing x= in CQ6 of 18 27

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

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

28. 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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