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.