1 Edge-bipancyclicity of star graphs under edge-fault tolerant Applied Mathematics and Computation, Volume 183, Issue 2, 15 December 2006, Pages Min Xu, Xiao-Dong Hu and Qiang Zhu 學 生 : 蕭 旻 昆 指導教授 : 洪 春 男

2 Abstract Introduction Star graphs Edge-pancyclicity of star graphs with edge-fault Outline

3 Abstract It has been shown by Li [T.-K. Li, Cycle embedding in star graphs with edge faults, Appl. Math. Comput. 167 (2005) 891–900] that S n contains a cycle of length from 6 to n! when the number of fault edges in the graph does not exceed n － 3. We improve this result by showing that for any edge subset F of S n with |F|≤n － 3 every edge of S n － F lies on a cycle of every even length from 6 to n! provided n≥3.

4 Introduction [7] showed that any cycle of even length from 6 to n! can be embedded into S n. Hsieh et al. [5] and Li et al. [9], proved that the n-dimensional star graph S n is (n-3)-edge-fault tolerant Hamiltonian laceable for n ≥ 4. Recently, Li [8] considered the edge-fault tolerance of star graphs and showed that cycles of even length from 6 to n! can be embedded into the n- dimensional star graphs when the number of the fault edges are less than n - 3.

5 Introduction In Section 2, we give the definition and basic properties of the n-dimensional star graph S n. In Section 3, we discuss the edge-fault-tolerant edge-bipancyclicity of the star graphs.

6 Star graphs Lemma 1 (Li [8]). There are n vertex-disjoint S n - 1’s in S n for n ≥ 2.

7 Star graphs let H i:j = (V i:j,E i:j ), where V i:j = {u ∈ V(S n ) ｜ u=u 1 u 2 …u i …u n,u i = j} and E i:j = {(u, v) ∈ E(S n )u,v ∈ V i:j } for 1 ≤j ≤ n. Then {V i:1,V i:2,...,V i : n } is a partition of V(S n ) and H i : j is isomorphic toS n － 1. we will use S i:j n-1 to denote the subgraph H i : j in the above partition. Specifically, we use S j n-1 as the abbreviation of S n:j n-1 and call it the jth (n-1)-dimensional subgraph of S n.

8 H i:j (2314 ， 1324 ， 2341 ， 4321 ， 4312 ， 1342)

9 For a vertex x,we use x i to denote the ith digits of vertex x. For a set of distinct edges e 1 =(x 1,y 1 ),e 2 =(x 2,y 2 ),..., e m =(x m,y m ) in S i n － 1 for n≥3 with x 1 1 ＝ x 2 1 ＝... ＝ x m 1 ＝ j and y 1 1 =y 2 1 =y m 1 =k, we call the edge set {e 1, e 2,..., e m } a set of (i,j,k)-edges. Obviously, a set of (i,j,k)- edges is also a set of (i,k,j)-edges since the edges we discuss in this paper have no direction. Star graphs

10 n=8 ， i=5 ， j=3 ， k=6 x 1 = y 1 = x 2 = y 2 = Star graphs

11 Theorem 2 The n-dimensional star graph S n is (n-3)-edge- fault Hamiltonian laceable for n≥4. Star graphs

12 Lemma 3 For any edge e and e’ of S 4, there exists a cycle of even length from 6 to 24 in S 4 e’containing e. Proof Since S n is edge-symmetric, without loss of generality, we assume that e=(1234,3214).By Theorem 2,there exists a Hamiltonian path P connecting the vertices 1234 and 3214 in S n -e’. Then P + e is a cycle of length 24 containing e in S 4 -e’. Let l be any even integer with 6≤l≤22.we need to construct a set of cycles of length l such that the intersection of their edge sets is e.

13 L = 6 L = 8 L = 10 L = 12 L = 14 L = 16 L = 18 L = 20 L = e

14 Lemma 4 For any edge e and an edge set F with |F| = n-3 and n ≥ 5, there exists a subgraph S i:j n- 1 =(V i:j,E i:j ) such that e ∈ E i : j and |E i : k ∩F ｜ ≤ n － 4 for all 1 ≤ k ≤ n.

15 Algorithm 5 Input m, n, where n ≥ 4 and 3 ≤ m ≤ n. Output (n-2) disjoint paths. 1. Select arbitrary (n-2)vertices, say a 1,a 2,..,a n-2 from V (S m n-1 ) with the first digits equal to 1 2. For i =1 to (n-2) 3. P i (a i )For j=2 to m 4. v =last Vertex(P i ) //Suppose that v = v 1 v 2 v n-1 v n. 5. //last Vertex(P i ) returns the last vertex in P i. 6. Append v n v 2..v n-1 v 1 to P i 7. Append v l v 2.. v l-1 v n v l+1.. v n-1 v 1 to P i where v l = j 8. Append v n v 2.. v n-1 v 1 to P i, where v is the last vertex in P i 9. Output P 1,P 2,...,P n-2

16 例 1: n=4 ， m= ， 3241 ， 2341 ， 1342 ， 3142 ， ， 3421 ， 4321 ， 1324 ， 3124 ， 4123 例 2: n=4 ， m= ， 4231 ， 2431 ， 1432 ， 3412 ， 2413 ， 4213 ， ， 4321 ， 2341 ， 1342 ， 3142 ， 2143 ， 4123 ， 3124

18 Lemma 6 There are (n-2) disjoint paths of length (2m-1) crossing S 1 n-1,S 2 n-1,..., S m n-1 such that the endpoints of these paths are in S m n-1 for n≥3 and 3 ≤ m ≤ n. The endpoints of these paths are adjacent for m = 3,4.

19 Edge-pancyclicity of star graphs with edge-fault Theorem 9 For any edge subset F of S n with |F| ≤ n-3 and any edge e ∈ S n -F, there exists a cycle of even length from 6 to n! in S n -F containing e provided n ≥ 3.

20 Case 1 6 ≤ l ≤ (n-1)! By the induction hypothesis, there exists a cycle C of length l in S an n-1 containing e.Specially, we use C1 and C2 to denote the cycles of length (n-1)!-2 and (n-1)!, respectively. Case 2 (n-1)! + 2 ≤ l ≤ 3(n-1)! In this case, we can write l = l 1 + l 2 + l 3 where l 1, l 2 and l 3 satisfy the following conditions: l 1 =(n-1)!-2 or (n-1)! l 2 =2 or 6 ≤ l 2 ≤ (n-1)! l 3 =2 or 6 ≤ l 3 ≤ (n-1)!

21 Case 3 3(n-1)! + 2 ≤ l ≤ n! In this case, we can write l = l 1 + l 2 + l 3 + … + l n where l 1 ≥ l 2 ≥ l 3 ≥…≥ l n and the following restrictions should be satisfied: l 1 = (n-1)!-2 or (n-1)! l 2 = (n-1)!-2 or (n-1)! l 3 = 2 or 6 ≤ l 3 ≤ (n-1)! l 4 = 0 or 2 or 6 ≤ l 3 ≤ (n-1)! … l n = 0 or 2 or 6 ≤ l n ≤ (n-1)!