1. Panconnectivity and Edge- Pancyclicity of 3-ary N-cubes 指導教授 : 黃鈴玲 老師 學生 : 郭俊宏 Sun-Yuan Hsieh, Tsong-Jie Lin and Hui-Ling Huang Journal of Supercomputing (accepted)

2. Outline Introduction Preliminaries Panconnectivity of 3-ary n-cubes Edge-pancyclicity of 3-ary n-cubes Concluding Remarks

3. Introduction The panconnectivity of the 3-ary n -cube Q n 3 : Given two arbitrary distinct nodes x and y in Q n 3, there exists an x - y path of length l ranging from n to 3 n − 1, where n is the diameter of Q n 3. Edge-pancyclicity of the 3-ary n -cube Q n 3 : Every edge in Q n 3 lies on a cycle of every length ranging from 3 to 3 n.

4. Preliminaries A graph G is said to be Hamiltonian if it contains a Hamiltonian cycle. G is Hamiltonian-connected if there exists a Hamiltonian path between every two distinct vertices of G. G is edge-pancyclic if every edge of G lies on a cycle of every length from 3 to | V(G) |.

5. Preliminaries The k -ary n -cube Q n k ( k ≥ 2 and n ≥ 1) has N = k n nodes each of the form x = x n x n −1... x 1, where 0 ≤ x i < k for all 1 ≤ i ≤ n. Two nodes x = x n x n −1... x 1 and y = y n y n −1... y 1 in Q n k are adjacent if and only if there exists an integer j, 1 ≤ j ≤ n, such that x j = y j ± 1 (mod k ) and x l = y l, for every l ∈ {1, 2,..., n } − { j } 012 ex. 011 010 002002 022022 112 212 When k=3

6. Preliminaries Each node has degree 2 n when k ≥ 3, and degree n when k = 2. In this paper, we pay our attention on k = 3. The i th position, from the right to the left, of the n -bit string x n x n −1... x 1, is called the i -dimension. We can partition Q n 3 along the i -dimension by regarding the graph comprised by 3 disjoint copies, Q n −1 3 [0], Q n −1 3 [1], and Q n −1 3 [2]. There are exactly 3 n −1 edges which form a perfect matching between Q n −1 3 [ j ] and Q n −1 3 [ j + 1], j ∈ {0, 1, 2}.

7. Q 2 3 [0] Q 2 3 [1] Q 2 3 [2] Q33Q33 0 1 2 0 12 1 2 0 010 011 012 i = 1

8. Panconnectivity of 3-ary n-cubes Lemma 1 [10] The k -ary n -cube is Hamiltonian-connected when k is odd. Lemma 2 For any two distinct nodes x, y ∈ V ( Q 2 3 ) and any integer l with 2 ≤ l ≤ 8, Q 2 3 contains an x - y path of length l.

9. Proof: We attempt to construct x - y paths of all lengths from 2 to 8. Case 1. x = 00 and y = 01

10. Case 2. x = 00 and y = 11

11. Theorem 1. For any two distinct nodes x, y ∈ V ( Q n 3 ) and any integer l with n ≤ l ≤ 3 n − 1, there exists an x - y path of length l. Proof: (by induction on n ) n = 1 : Q 1 3 is isomorphic to C 3. n = 2 : hold by Lemma 2 Suppose that the result holds for Q n −1 3. Consider Q n 3 : Partition Q n 3 along the dimension i ( for some i) into three subcubes Q n −1 3 [0], Q n−1 3 [1], and Q n−1 3 [2]. There are the following two scenarios.

12. Case 1. x and y are in the same subcubes. WLOG, assume x, y  V ( Q n − 1 3 [0] ). We now attempt to construct an x - y path of every length l with n ≤ l ≤ 3 n − 1. Subcase 1.1. n ≤ l ≤ 3 n −1 − 1 x y Q n -1 3 [0] Q n -1 3 [1] Q n -1 3 [2]

13. Subcase 1.2. 3 n −1 ≤ l ≤ 2 · 3 n −1 − 1. x y Q n -1 3 [0] Q n -1 3 [1] Q n -1 3 [2] P0P0 P1P1 u v u’ v’ P 0 [ x, y ] of length l 0 with 3 n −1 − n ≤ l 0 ≤ 3 n −1 − 1. P 1 [ u ’, v ’] of length l 1 with n − 1 ≤ l 1 ≤3 n −1 − 1.

14. x y Q n -1 3 [0] Q n -1 3 [1] Q n -1 3 [2] P0P0 P1P1 u v u’ v’ Case 1.3. 2 · 3 n −1 ≤ l ≤ 3 n − 1. w w’ P2P2 path P 0 [ x, y ] of length l 0 with 3 n −1 − n ≤ l 0 ≤ 3 n −1 −1. path P 1 [ u ’, w ] of length l 1 with n − 1 ≤ l 1 ≤ 3 n −1 − 1. Hamiltonian path P 2 [ w ’, v ’] of length l 2 = 3 n −1 − 1.

15. Case 2. x and y are in different subcubes. WLOG, assume x  V ( Q n −1 3 [0]) and y  V ( Q n −1 3 [1]). Subcase 2.1. n ≤ l ≤ 3 n −1 − 1. x Q n -1 3 [0] Q n -1 3 [1] Q n -1 3 [2] u1u1 y P1P1 If u 1 = y, then we can partition Q n 3 along another dimension i ’(  i ) such that x and y are in the same subcube.. Thus we assume u 1  y. path P 1 [ u 1, y ] of length l 1 with n − 1 ≤ l 1 ≤ 3 n −1 − 2.

16. Case 2.2. 3 n −1 ≤ l ≤ 3 n − 1 x y Q n -1 3 [0] Q n -1 3 [1] Q n -1 3 [2] P0P0 P1P1 v u1u1 v2v2 u2u2 P2P2 path P 1 [ u 1, y ] of length l 1 with 3 n −1 −2 n ≤ l 1 ≤ 3 n −1 − 1 P 2 [ v 2, u 2 ] of length l 2 with n − 1 ≤ l 2 ≤ 3 n −1 − 1. path P 0 [ x, v ] of length l 0 with n − 1 ≤ l 0 ≤ 3 n −1 − 1.

17. 4 Edge-pancyclicity of 3-ary n-cubes Lemma 3 For any edge ( x, y ) ∈ E (Q 2 3 ) and any integer l with 3 ≤ l ≤ 9, there exists a cycle C of length l such that ( x, y ) is in C. Proof: Due to the structure property of Q 2 3, we only need to consider the edge (00, 01).

18. Theorem 2 For any edge ( x, y ) ∈ E ( Q n 3 ), and any integer l with 3 ≤ l ≤ 3 n, there exists a cycle C of length l such that ( x, y ) is in C. That is, Q n 3 is edge-pancyclic. Proof: (by induction on n) n = 1 : Q 1 3 is isomorphic to C 3. n = 2 : hold by Lemma 2 Suppose that the result holds for Q n−1 3. Consider Q n 3 : Partition Q n 3 along the dimension i (for some i) into three subcubes Q n−1 3 [0], Q n−1 3 [1], and Q n−1 3 [2].

19. Case 1. 3 ≤ l ≤ 3 n−1. x y Q n-1 3 [0] Q n-1 3 [1] Q n-1 3 [2]

20. Case 2. 3 n−1 + 1 ≤ l ≤ 3 n. x y Q n-1 3 [0] Q n-1 3 [1] Q n-1 3 [2] p0p0 p1p1 v u1u1 v2v2 u2u2 p2p2 v1v1 C0C0 Q n−1 3 [0] contains a cycle C 0 of length 3 n−1 such that (x, y ) is in C 0. path P 0 [ x, v ] = from C 0 whose length l 0 satisfies 3 n−1 − 2 n ≤ l 0 ≤ 3 n−1 − 1. P 1 [ u 1, v 1 ] of length l 1 with n − 1 ≤ l 1 ≤ 3 n−1 − 1. P 2 [ u 2, v 2 ] of length l 2 with n − 1 ≤ l 2 ≤ 3 n−1 − 1

21. Concluding Remarks In this paper, we have focused on fault- tolerant embedding, where a 3-ary n -cube acts as the host graph and paths (cycles) represent the guest graphs. A future work is to extend our result to the k -ary n-cube for k > 3.