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

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

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. Introduction

S4S4 hamiltonian laceable hyper hamiltonian laceable strongly hamiltonian laceable

S 4 star graph

Definition and basic properties Definition 1. The n-dimensional star graph is denoted by S n.The vertex set V of S n is {a 1... a n |a 1... a n is a permutation of 1,2,…n} and the edge set E is {(a 1 a 2... a i-1 a i a i+1... a n,a i a 2... a i-1 a 1 a i+1... a n )|a 1... a n ∈ V and 2  i  n}.

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

Proposition 2. Given k with 1  k  n-1 and b k+1... b n, a vertex u = u 1... u n is adjacent to S k bk+1...bn if and only if u k+1... u i-1 u 1 u i+1... u n = b k+1 b k+2... b n for some i with k+1  i  n. Definition and basic properties

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

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

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

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

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

Main result Case 2. j 1 = j 2 = j. The proof of this case is similar to that of case 2 in Theorem 1 except that the path in S j n-1 from x to y is of length (n-1)!-2. S j n-1 xy u1 S j3 n-1

Main result Theorem 3. S n is (n-4)-edge fault tolerant hyper hamiltonian laceable for n>=4. Case 1. v, x, y are in the same substar, say S j1 n-1.

Main result Case 2. v,x  S j1 n-1 and y  S j2 n-1 with j 1 ≠ j 2.

Main result Case 3. v  S j1 n-1 and x, y  S j2 n-1 with j 1 ≠j 2.

Main result Case 4. v  S j1 n-1, x  S j2 n-1, and y  S j3 n-1 for distinct j 1, j 2, and j 3.

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