Hyper hamiltonian laceability on edge fault star graph

1. Hyper hamiltonian laceability on edge fault star graph Tseng-Kuei Li , Jimmy J.M. Tan , Lih-Hsing Hsu Information Sciences 指導教授: 洪 春 男 老師 學 生 : 蕭 旻 昆

2. Outline ‧Introduction ‧Definition and basic properties ‧Main result ‧Conclusion

3. Introduction n-dimensional star graph is (n-3)-edge fault tolerant hamiltonian laceable, (n-3)-edge fault tolerant strongly hamiltonian laceable, (n-4)-edge fault tolerant hyper hamiltonian laceable.

4. 1234 4231 3241 3214 2134 2431 1 4 2314 3124 3421 2341 1324 4321 3412 2413 4213 1432 1423 4312 2 3 4123 1342 4132 1243 3142 2143 hamiltonian laceable hyper hamiltonian laceable strongly hamiltonian laceable S4

5. 1234 4231 3241 2134 2431 3214 4 1 3421 2314 3124 2341 1324 4321 2413 3412 1432 1423 4213 4312 3 2 4123 1342 4132 1243 3142 2143 S4 star graph

6. Definition and basic properties Definition 1. The n-dimensional star graph is denoted by Sn.The vertex set V of Sn is {a1 . . . an|a1 . . . an is a permutation of 1,2,…n} and the edge set E is {(a1a2 . . . ai-1aiai+1 . . . an,aia2 . . . ai-1a1ai+1 . . . an)|a1 . . . an∈ V and 2 i n}.

7. Definition and basic properties Proposition 1. There are n!/k! vertex-disjoint Sk’s embedded in Sn for k>=1.

8. Definition and basic properties Proposition 2. Given k with 1 k n-1 and bk+1 . . . bn, a vertex u = u1 . . . un is adjacent to Skbk+1...bn if and only if uk+1 . . . ui-1u1ui+1 . . . un = bk+1bk+2 . . . bn for some i with k+1 i  n.

9. Definition and basic properties Corollary 1. There are (k-1)! edges between Skbk+1...bn and Skb’k+t1...b’n if there is exactly one different bit between bk+1 . . . bn and b’k+1 . . . b’n.

10. Main result ‧edge fault tolerant hamiltonian laceability (eftHL) ‧edge fault tolerant strongly hamiltonian laceability (eftSHL) ‧edge fault tolerant hyper hamiltonian laceability (eftHHL)

11. Main result Theorem 1. Sn is (n-3)-edge fault tolerant hamiltonian laceable for n  4. Case 1. j1≠j2. Let V ={1,2,…,n}. Since |F|  n-3 < (n-2)!/2 for n  5, Ei,j(S)∩F < (n-2)!/2 for any i ≠ ∈ V .

12. Main result Case 2. j1= j2 = j. There is a hamiltonian path P of Sjn-1 from x to y. The length of P is (n-1)!-1.

13. Main result Theorem 2. Sn is (n-3)-edge fault tolerant strongly hamiltonian laceable for n>=4. Case 1. j1≠j2. Let Ve be the number of vertices which are in the different partite set from x and which are not adjacent to Sj2n-1.

14. Main result Case 2. j1 = j2 = j. The proof of this case is similar to that of case 2 in Theorem 1 except that the path in Sjn-1 from x to y is of length (n-1)!-2. S jn-1 x y u1 S j3n-1

15. Main result Theorem 3. Sn is (n-4)-edge fault tolerant hyper hamiltonian laceable for n>=4. Case 1. v, x , y are in the same substar, say Sj1n-1.

16. Main result Case 2. v,x  Sj1n-1 and ySj2n-1 with j1≠ j2 .

17. Main result Case 3. v  Sj1n-1 and x , y  Sj2n-1 with j1≠j2.

18. Main result Case 4. v  Sj1n-1, xSj2n-1, and y Sj3n-1 for distinct j1, j2, and j3.

19. Conclusion ‧(n-3)-edge fault tolerant hamiltonian laceable, ‧(n-3)-edge fault tolerant strongly hamiltonian laceable ‧(n-4)-edge fault tolerant hyper hamiltonian laceable