Panconnectivity and Edge-Pancyclicity of 3-ary N -cubes

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

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 Qn3: Given two arbitrary distinct nodes x and y in Qn3, there exists an x-y path of length l ranging from n to 3n − 1, where n is the diameter of Qn3. • Edge-pancyclicity of the 3-ary n-cube Qn3: Every edge in Qn3 lies on a cycle of every length ranging from 3 to 3n.

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 Qnk(k ≥ 2 and n ≥ 1) has N = kn nodes each of the form x = xnxn−1 . . . x1, where 0 ≤ xi < k for all 1 ≤ i ≤ n. • Two nodes x = xnxn−1 . . . x1 and y =ynyn−1 . . . y1 in Qnk are adjacent if and only if there exists an integer j, 1 ≤ j ≤ n, such that xj = yj ± 1 (mod k) and xl = yl, for every l ∈ {1, 2, ..., n} − { j } 022 002 ex. 010 112 011 When k=3 212 012

6. Preliminaries • Each node has degree 2n when k ≥ 3, and degree n when k = 2. In this paper, we pay our attention on k = 3. • The ith position, from the right to the left, of the n-bit string xnxn−1 . . . x1, is called the i-dimension. • We can partition Qn3 along the i-dimension by regarding the graph comprised by 3 disjoint copies, Qn−13[0], Qn−13[1], and Qn−13[2]. • There are exactly 3n−1 edges which form a perfect matching between Qn−13[j] and Qn−13[j + 1], j ∈ {0, 1, 2}.

7. Q33 i = 1 1 2 0 0 0 1 2 1 011 010 012 2 Q23[0] Q23[1] Q23[2]

8. Panconnectivity of 3-ary n-cubes • Lemma 1 [10] The k-ary n-cube is Hamiltonian-connected when k is odd. • Lemma 2For any two distinct nodes x, y ∈ V(Q23) and any integer l with 2 ≤ l ≤ 8, Q23 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 (Qn3) and any integer l with n ≤ l ≤ 3n − 1, there exists an x-y path of length l. Proof: (by induction on n) n = 1 : Q13 is isomorphic toC3. n = 2 : hold by Lemma 2 Suppose that the result holds for Qn−13. Consider Qn3: Partition Qn3 along the dimension i (for some i)into three subcubes Qn−13[0], Qn−13[1], and Qn−13[2]. There are the following two scenarios.

12. Case 1. x and y are in the same subcubes. WLOG, assume x,yV(Qn−13[0]). We now attempt to construct an x-y path of every length l with n ≤ l ≤ 3n − 1. Subcase 1.1. n ≤ l ≤ 3n−1 − 1 <induction hypothesis> x y Qn-13[0] Qn-13[1] Qn-13[2]

13. Subcase 1.2. 3n−1 ≤ l ≤ 2 · 3n−1 − 1. P0[x, y] of length l0 with 3n−1−n ≤ l0 ≤ 3n−1 − 1. <induction hypothesis> P1[u’, v’] of length l1 with n − 1 ≤ l1 ≤3n−1 − 1. <induction hypothesis> x u u’ P0 P1 v’ v y Qn-13[0] Qn-13[1] Qn-13[2]

14. Case 1.3.2 · 3n−1 ≤ l ≤ 3n − 1. path P0[x, y] of length l0 with 3n−1−n ≤ l0 ≤ 3n−1−1. <induction hypothesis> x w’ u u’ P2 P0 P1 path P1[u’,w] of length l1 with n − 1 ≤ l1 ≤ 3n−1 − 1. <induction hypothesis> v v’ w y Qn-13[0] Qn-13[1] Qn-13[2] Hamiltonian path P2[w’, v’] of length l2 = 3n−1 − 1. <Lemma 1>

15. Case 2. x and y are in different subcubes.WLOG, assume xV(Qn−13[0]) and yV(Qn−13[1]). Subcase 2.1.n ≤ l ≤ 3n−1 − 1. If u1 = y, then we can partition Qn3 along another dimension i’(i) such that x and y are in the same subcube. <Case 1>. x u1 P1 y Thus we assume u1y. path P1[u1, y] of length l1 with n − 1 ≤ l1 ≤ 3n−1 − 2. <induction hypothesis> Qn-13[0] Qn-13[1] Qn-13[2]

16. Case 2.2.3n−1 ≤ l ≤ 3n − 1 path P0[x, v] of length l0 withn − 1 ≤ l0 ≤ 3n−1 − 1. <induction hypothesis> x u2 u1 P1 P2 P0 y path P1[u1, y] of length l1 with 3n−1 −2n ≤ l1 ≤ 3n−1 − 1 v2 v Qn-13[0] Qn-13[1] Qn-13[2] P2[v2, u2] of length l2 with n − 1 ≤ l2 ≤ 3n−1 − 1.

17. 4 Edge-pancyclicity of 3-ary n-cubes Lemma 3For any edge (x, y) ∈ E(Q23) 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 Q23, we only need to consider the edge (00, 01).

18. Theorem 2 For any edge (x, y) ∈ E(Qn3), and any integer l with 3 ≤ l ≤ 3n, there exists a cycle C of length l such that (x, y) is in C. That is, Qn3 is edge-pancyclic. Proof:(by induction on n) n = 1 : Q13 is isomorphic to C3. n = 2 : hold by Lemma 2 Suppose that the result holds for Qn−13. Consider Qn3: Partition Qn3 along the dimension i (for some i) into three subcubes Qn−13[0], Qn−13[1], and Qn−13[2].

19. Case 1. 3 ≤ l ≤ 3n−1. < induction hypothesis> x y Qn-13[0] Qn-13[1] Qn-13[2]

20. Case 2. 3n−1 + 1 ≤ l ≤ 3n. Qn−13[0] contains a cycle C0 of length 3n−1 such that (x, y) is in C0. path P0[x, v] = <x, y, ..., v> from C0 whose length l0 satisfies 3n−1 − 2n ≤ l0 ≤ 3n−1 − 1. <induction hypothesis> u2 x u1 y p1 p0 p2 P1[u1, v1] of length l1 with n − 1 ≤ l1 ≤ 3n−1 − 1. C0 v1 v2 v P2[u2, v2] of length l2 with n − 1 ≤ l2 ≤ 3n−1 − 1 Qn-13[0] Qn-13[1] Qn-13[2]

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.