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)

Outline

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

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 Qn

3.

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.

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) |.

Preliminaries The k-ary n-cube Qn

k (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 Qn

k 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 }

012

ex.

011

010002 022

112

212 When k=3

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−1

3[0], Qn−13[1], and Qn−1

3[2]. There are exactly 3n−1 edges which form a perfect

matching between Qn−13[j] and Qn−1

3[j + 1], j {0, 1, ∈2}.

Q23[0] Q2

3[1] Q23[2]

Q33

0

1

2

0 1 2

1

2

0

010 011 012

i = 1

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(Q2

3) and any integer l with 2 ≤ l ≤ 8, Q2

3 contains an x-y path of length l.

Proof: We attempt to construct x-y paths of all lengths from 2 to 8.

Case 1. x = 00 and y = 01

Case 2. x = 00 and y = 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 : Q1

3 is isomorphic to C3. n = 2 : hold by Lemma 2 Suppose that the result holds for Qn−1

3. Consider Qn

3: Partition Qn

3 along the dimension i (for some i) into three subcubes Qn−1

3[0], Qn−13[1], and Qn−1

3[2]. There are the following two scenarios.

Case 1. x and y are in the same subcubes. WLOG, assume x,y V(Qn−1

3[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

x

y

Qn-13[0] Qn-1

3[1] Qn-13[2]

<induction hypothesis>

Subcase 1.2. 3n−1 ≤ l ≤ 2 · 3n−1 − 1.

x

y

Qn-13[0] Qn-1

3[1] Qn-13[2]

P0 P1

u

v

u’

v’

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

y

Qn-13[0] Qn-1

3[1] Qn-13[2]

P0 P1

u

v

u’

v’

Case 1.3. 2 · 3n−1 ≤ l ≤ 3n − 1.

w

w’

P2

path P0[x, y] of length l0 with 3n−1−n ≤ l0 ≤ 3n−1−1.<induction hypothesis>

path P1[u’,w] of length l1 with n − 1 ≤ l1 ≤ 3n−1 − 1.

<induction hypothesis>

Hamiltonian path P2[w’, v’] of length l2 = 3n−1 − 1.

<Lemma 1>

Case 2. x and y are in different subcubes. WLOG, assume xV(Qn−1

3[0]) and yV(Qn−13[1]).

Subcase 2.1. n ≤ l ≤ 3n−1 − 1.

x

Qn-13[0] Qn-1

3[1] Qn-13[2]

u1

y

P1

If u1 = y, then we can partition Qn

3 along another dimension i’( i) such that x and y are in the same subcube. <Case 1>.

Thus we assume u1 y. path P1[u1, y] of length l1 with n − 1 ≤ l1 ≤ 3n−1 − 2.<induction hypothesis>

Case 2.2. 3n−1 ≤ l ≤ 3n − 1

x

y

Qn-13[0] Qn-1

3[1] Qn-13[2]

P0

P1

v

u1

v2

u2

P2

path P1[u1, y] of length l1 with 3n−1 −2n ≤ l1 ≤ 3n−1 − 1

P2[v2, u2] of length l2 with n − 1 ≤ l2 ≤ 3n−1 − 1.

path P0[x, v] of length l0 with n − 1 ≤ l0 ≤ 3n−1 − 1.

<induction hypothesis>

4 Edge-pancyclicity of 3-ary n-cubesLemma 3

For 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 Q2

3, we only need to consider the edge (00, 01).

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, Qn

3 is edge-pancyclic.

Proof: (by induction on n) n = 1 : Q1

3 is isomorphic to C3. n = 2 : hold by Lemma 2 Suppose that the result holds for Qn−1

3. Consider Qn

3: Partition Qn

3 along the dimension i (for some i) into three subcubes Qn−1

3[0], Qn−13[1], and Qn−1

3[2].

Case 1. 3 ≤ l ≤ 3n−1.

x

y

Qn-13[0] Qn-1

3[1] Qn-13[2]

< induction hypothesis >

Case 2. 3n−1 + 1 ≤ l ≤ 3n.

x

y

Qn-13[0] Qn-1

3[1] Qn-13[2]

p0p1

v

u1

v2

u2

p2

v1C0

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>P1[u1, v1] of length l1 with n − 1 ≤ l1 ≤ 3n−1 − 1.

P2[u2, v2] of length l2 with n − 1 ≤ l2 ≤ 3n−1 − 1

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.