The Mutually Independent Bipanconnected Property
for Hypercube
Yuan-Kang Shih
Department of Computer Science, National Chiao Tung University,
Hsinchu, Taiwan 30010, R.O.C. Email: [email protected]
Jimmy J. M. Tan
Department of Computer Science, National Chiao Tung University,
Hsinchu, Taiwan 30010, R.O.C. Email: [email protected]
Lih-Hsing Hsu
Department of Computer Science and Information Engineering
Providence University, Taichung, Taiwan 43301, R.O.C.
Email: [email protected]
Abstract—A graph is denoted by G with the vertex set V (G) and the edge set E(G). A path P = v0, v1, · · · , vm is a sequence of adjacent vertices. Two paths with equal length P1 = u1, u2, . . . , um and P2 = v1, v2, . . . , vm from a to b are independent if u1= v1 = a, um= vm= b, and ui= vi for 2 ≤ i ≤ m − 1. Paths with equal length {Pi}n
i=1 from a to b are mutually independent if they are pairwisely independent. Let u and v be two distinct vertices of a bipartite graph G, and let l be a positive integer length, dG(u, v) ≤ l ≤ |V (G) − 1| with (l − dG(u, v)) being even. We say that the pair of vertices u, v is (m, l)-mutually independent bipanconnected if there exist m mutually independent paths {Pil}mi=1 with length l from u to v. In this paper, we explore yet another strong property of the hypercubes. We prove that every pair of vertices u and v in the n-dimensional hypercube, with dQn(u, v) ≥ n − 1, is (n − 1, l)-mutually independent bipanconnected for every l, dQn(u, v) ≤ l ≤ |V (Qn) − 1| with (l − dQn(u, v)) being even. As for dQn(u, v) ≤ n − 2, it is also (n − 1, l)-mutually independent bipanconnected if l ≥ dQn(u, v) + 2, and is only (l, l)-mutually independent bipanconnected if l = dQn(u, v).
I. INTRODUCTION
For the graph definitions and notations we refer the reader to [6]. A graph is denoted by G with the vertex set V (G) and the edge set E(G). The simulation of one architecture by another is an important issue in interconnection networks. The problem of simulating one network by another is also called embedding problem. One particular problem of path embedding deals with finding all the possible length of paths in an interconnection network.
A path P = v0, v1, . . . , vm is a sequence of adjacent
vertices. We also writeP = v0, . . . , vi, Q, vj, . . . , vm where
Q is a path vi, . . . , vj. A cycle C = v0, v1, . . . , vm, v0 is a
sequence of adjacent vertices where the first vertex is the same as the last one. The length of a pathP (a cycle C respectively) is the number of edges inP (in C respectively).
A cycle of G is a hamiltonian cycle if it traverses all the vertices exactly once. A graphG is called a hamiltonian graph ifG contains a hamiltonian cycle. A path of G is a hamitonian path if it contains all the vertices exactly once. A graph G is hamiltonian connected if there exists a hamiltonian path between any two different vertices of G. A graph G = (B ∪ W, E) is bipartite if V (G) is the union of two disjoint sets B and W such that every edge joins B with W . It is easy to see that any bipartite graph with at least three vertices is
not hamiltonian connected. A bipartite graphG is hamiltonian laceable if there exists a hamiltonian path joining any two vertices from different partite sets. A graphG is pancyclic [2] ifG includes cycles of all lengths. If these cycles are restricted to even length,G is called a bipancyclic graph. The distance from x to y, written dG(x, y), is the least length among all
paths from x to y in G. A graph is panconnected if, for any two different vertices x and y, there exists a path of length l joining x and y, for every l, dG(x, y) ≤ l ≤ |V (G)| − 1.
The concept of panconnected graphs is proposed by Alavi and Williamson [1]. It is not hard to see that any bipartite graph with at least 3 vertices is not panconnected. Therefore, the concept of bipanconnected graphs is proposed. A bipartite graph is bipanconnected if, for any two different vertices x andy, there exists a path of length l joining from x to y, for every l, dG(x, y) ≤ l ≤ |V (G)| − 1 and (l − dG(x, y)) being
even. There are many studies on bipanconnected graphs and bipancyclic graphs [3], [7], [9], [13].
We introduce some terms defined recently. Two pathsP1=
u1, u2, . . . , um and P2 = v1, v2, . . . , vm from a to b are
independent [10] if u1= v1= a, um= vm= b, and ui= vi
for 2 ≤ i ≤ m − 1. Paths with equal length {Pi}ni=1 from
a to b are mutually independent [10] if they are pairwisely independent. Two cycles C1 = u1, u2, . . . , um, u1 and
C2 = v1, v2, . . . , vm, v1 beginning at x are independent if
u1 = v1 = x and ui = vi for 2 ≤ i ≤ m. Cycles with
equal length{Ci}ni=1beginning atx are mutually independent
if every two cycles are independent. Two hamiltonian paths P1 = u1, u2, . . . , u|V (G)| and P2 = v1, v2, . . . , v|V (G)|
are independent beginning at x [5] if u1 = v1 = x and
ui= vi for2 ≤ i ≤ |V (G)|, denoted P1: x → u|V (G)| and
P2 : x → v|V (G)|. Hamiltonian paths {Pi}ni=1 are mutually
independent hamiltonian paths beginning at x [5] if any two of them are independent beginning atx.
An n-dimensional hypercube, denoted by Qn, is a graph
with2n vertices, and each vertexu can be distinctly labeled
by an n-bit binary string, u = un−1un−2...u1u0. There is an
edge between two vertices if and only if their binary labels differ in exactly one bit position. Let(u, v) be an edge in Qn.
If the binary labels of u and v differ in ith position, then the edge between them is said to be inith dimension and the edge (u, v) is called an ith dimension edge. We use Q0
n−1to denote
2009 10th International Symposium on Pervasive Systems, Algorithms, and Networks
978-0-7695-3908-9/09 $26.00 © 2009 IEEE DOI 10.1109/I-SPAN.2009.88
the subgraph of Qn induce by {u ∈ V (Qn) | ui = 0} and
Q1n−1to denote the subgraph ofQninduced by{u ∈ V (Qn) |
ui = 1}. Q0n−1 and Q1n−1 are all isomorphic to Qn−1.Qn
can be decomposed into Q0n−1 and Q1n−1 by dimension i, andQ0n−1andQ1n−1are(n− 1)-dimensional subcubes of Qn
induced by the vertices with theith bit position being 0 and 1 respectively. For each vertex u in Qi
n−1,i = {0, 1}, there
is exactly one vertex inQi−1n−1, denoted by¯u, such that (u, ¯u) is an edge inQn. There are many studies on the hypercubes
[5], [9], [11], [12], [14], [15].
We now introduce a new concept. Let u and v be two distinct vertices of a bipartite graphG and let l be a positive integer length,dG(u, v) ≤ l ≤ |V (G) − 1| with (l − dG(u, v))
being even. We say that the pair of vertices u, v is (m, l)-mutually independent bipanconnected if there exist m mutu-ally independent paths {Pil}mi=1 with length l from u to v. In this paper,we explore yet another strong property of the hypercubes. We prove that every pair of vertices u and v in the n-dimensional hypercube, with dQn(u, v) ≥ n − 1, is (n − 1, l)-mutually independent bipanconnected for every l, dQn(u, v) ≤ l ≤ |V (Qn) − 1| with (l − dQn(u, v)) being even. As fordQn(u, v) ≤ n − 2, it is also (n − 1, l)-mutually independent bipanconnected ifl ≥ dQn(u, v) + 2, and is only (l, l)-mutually independent bipanconnected if l = dQn(u, v). Our result strengthens a previous results of Sun et al. [14], and Li et al. [9]. Li et al. [9] proved that the hypercubeQn is
bipanconnected forn ≥ 2. Sun et al. [14] proved that there are n − 1 mutually independent hamiltonian paths in Qn between
every two vertices from different partite sets forn ≥ 4. The number “n − 1” in our result is tight as we have the following observation. Because each vertex of the hypercube Qn has
exactlyn edges incident with it, we can expect at most n − 1 mutually independent paths when the given two vertices are adjacent.
II. PRELIMINARIES
In order to prove our claim, we need some previous results. The following results state that there exist n − 1 mutually independent hamiltonian paths between two vertices. We shall strengthen the result by showing that there existn−1 mutually independent paths of lengthl between two vertices, for every reasonable lengthl.
Theorem 1. [14] Let x and y be two vertices from different partite sets of Qn, for n ≥ 4. Then there exist n − 1 mutually
independent hamiltonian paths joining x to y.
Theorem 2. [14] For n ≥ 4, there are n mutually independent hamiltonian cycles beginning at any vertex x in Qn.
A hamiltonian laceable graphG is hyper hamiltonian lace-able if for any vertex u, there is a hamiltonian path of G−{u} between every pair of vertices in the opposite partite set ofu. Theorem 3. [8] For n ≥ 2, the hypercube Qn is hyper
hamiltonian laceable.
Lemma 1. [4] Let Fv be a set of faulty vertices in Qn. For
n ≥ 3, if |Fv| ≤ n − 2, there exists a path of Qn− Fv with
any odd length l, 3 ≤ l ≤ 2n− 2|F
v| − 1, between any two
adjacent vertices.
Lemma 2. [14] Qn−{x, y} is hamiltonian laceable, if x and
y are any two vertices from different partite sets of Qn with
n ≥ 4.
Lemma 3. [5] In Qn, n ≥ 2, let u be any vertex, and v1,
v2,. . . , vn−1be any n−1 vertices in the opposite partite set of
u. There exist n − 1 mutually independent hamiltonian paths beginning at u of Qn such that{Pi: u → vi}n−1i=1.
III. MUTUALLYINDEPENDENTBIPANCONNECTED
PROPERTY OFHYPERCUBE
Lemma 4. Let x and y be two vertices from different partite sets of Qnwith n ≥ 4. There exists a path of every odd length
from 1 to 2n− 3 joining any two adjacent fault-free vertices
in Qn− {x, y}.
Proof: Let u, v be two adjacent fault-free vertices in Qn−
{x, y}. Because u and v are adjacent fault-free vertices, there exists a path of length 1 joining from u to v in Qn− {x, y}. According to Lemma 1, there exists a path of every odd length from3 to 2n−2|2|−1(= 2n−5) joining u to v in Q
n−{x, y}.
Then by Lemma 2, there exists a path of length2n−3 joining
u to v in Qn− {x, y}. Therefore, the lemma holds.
Sun et al. [14] proved that any two hamiltonian path con-necting000 and 100 in Q3are not independent, in other words,
there do not exist 2 mutually independent hamiltonian paths in Q3 between 000 and 100. So, we will prove our theorem
beginning fromn ≥ 4 for Qn. We found that there are only
d mutually independent paths with length d if dQn(u, v) = d. In order to see this, we have the following lemma.
Lemma 5. Let u and v be two vertices of Qn with
dQn(u, v) = d, there are d and at most d mutually independent paths with length d joining from u to v.
Proof: By the symmetric property of the hypercubes, we may assume thatu is the vertex with n bits containing n 0’s, and v is the vertex with n bits containing d 1’s. In order to see the basic idea, we first give an example n = 6. In Q6. Letu = 000000 and v = 001111 then dQ6(u, v) = 4. We can
construct4 mutually independent paths with length 4 between u and v.
P0= u, 000001, 000011, 000111, v,
P1= u, 000010, 000110, 001110, v,
P2= u, 000100, 001100, 001101, v, and
P3= u, 001000, 001001, 001011, v.
For generaln, let u = 0 · · · 0 = 0n andv = 0 · · · 01 · · · 1 =
0n−d1d, thend Qn(u, v) = d. P0= 0n, 0n−11, 0n−212, · · · , 0n−d+11d−1, 0n−d1d, P1= 0n, 0n−210, 0n−3120, · · · , 0n−d1d−201, 0n−d1d, P2= 0n, 0n−3102, 0n−41202, · · · , 0n−d1d−3012, 0n−d1d, P3= 0n, 0n−4103, 0n−51203, · · · , 0n−d1d−4013, 0n−d1d, .. . Pd−2= 0n, 0n−d−110d−2, 0n−d120d−2, 0n−d120d−31, 0n−d120d−412· · · , 0n−d1201d−3, 0n−d1d, Pd−1= 0n, 0n−d10d−1, 0n−d10d−21, 0n−d10d−312, 0n−d10d−413, · · · , 0n−d101d−2, 0n−d1d.
{P0, P1, . . . , Pd−1} form d mutually independent paths with
lengthd joining u to v. If there exists a (d+1)th path Pwith lengthd between u and v such that Pis mutually independent to the firstd paths. So the first vertex after the beginning vertex u of P has to be different from all those of Pi i = 0 to
d − 1. Without loss of generality, assume that the first vertex after the beginning vertex u of P is (x)i = 0i10n−i−1 for 0 ≤ i ≤ n − d − 1. It is easy to see that dQn((x)i, v) = d + 1, since there ared+1 distinct bits between (x)iandv. Therefore,
it is impossible to find out a (d + 1)th path with length d between u and v which is independent to P0, P1, . . . , Pd−1.
We now show our main result Theorem 5 below. Our proof is by induction onn, for Qn. The base case isn = 4.
Theorem 4. Let u and v be a pair of vertices of Q4. If
dQ4(u, v) ≥ 3, Q4 is(3, l)-mutually independent
bipancon-nected for every l, dQ4(u, v) ≤ l ≤ 24−1 with (l−dQ4(u, v))
being even. As for dQ4(u, v) ≤ 2, it is also (3, l)-mutually
independent bipanconnected if l ≥ dQ4(u, v) + 2, and is only
(l, l)-mutually independent bipanconnected if l = dQ4(u, v).
We will use the notationPk
i orRki to denote a pathi with
lengthk.
Lemma 6. Let u and v be two adjacent vertices of Qn for
n ≥ 4. There exist n − 1 mutually independent paths {Pl i}n−1i=1
of Qnwith any odd length l, 3 ≤ l ≤ 2n− 1, joining from u
to v.
Proof: We choose a dimension to divide the hypercube Qn into two subcubes Q0n−1 and Q1n−1 such that u is a
black vertex inQ0n−1 andv a white vertex in Q1n−1. Notice
that ¯u = v. According to Theorem 2, there exist n − 1 mutually independent hamiltonian cycles {Ci}n−1i=1 in Q0n−1
beginning at u. For each k, 1 ≤ k ≤ 2n−1 − 1, let
Ci= u, Rki, xi,k, xi,k+1, . . . , xi,2n−1−1, u for 1 ≤ i ≤ n−1, where Rk
i = u, xi,1, xi,2, . . . , xi,k and |Rki| = k. Let
Sk
i = ¯xi,k, . . . , ¯xi,2, ¯xi,1, ¯u for 1 ≤ i ≤ n − 1. Combine
Rk
i and Ski, we let Pi2k+1 = u, Rik, xi,k, ¯xi,k, Ski, ¯u = v,
1 ≤ k ≤ 2n−1− 1, for 1 ≤ i ≤ n − 1. Then P2k+1
i is a path
joiningu to v with length 2k + 1. Since 1 ≤ k ≤ 2n−1− 1 so 3 ≤ 2k + 1 ≤ 2n − 1. Therefore, there exist n − 1
mutually independent paths {Pl
i}n−1i=1 with any odd length l,
3 ≤ l ≤ 2n− 1, joining from u to v.
Lemma 7. Let u and v be two vertices from the same partite set of Qn for n ≥ 4. There exist n − 1 mutually independent
paths{Pl
i}n−1i=1 of Qnwith any even length l, dQn(u, v)+2 ≤ l ≤ 2n− 2, joining from u to v.
Proof: We prove the statement by induction on n. By Theorem 4, the statement holds for n = 4. Suppose that the result holds for Qn−1, n ≥ 5. Without loss of generality,
let u and v be two black vertices of Qn. We may choose
a dimension to divide the hypercube Qn into two subcubes
Q0n−1 and Q1n−1 such that u ∈ Q0n−1 and v ∈ Q1n−1. Therefore,¯u and ¯v are two white vertices in Q1n−1andQ0n−1,
respectively. Assume that dQn(u, v) = d and d is even, then it is easy to see that dQn(u, ¯v) = dQn(u, v) − 1 = d − 1. According to the length l of the paths, we divide the proof into the following three cases. In each case, the length l is assumed to be an even number. We shall findn − 1 mutually independent paths with lengthl joining from u to v.
Case 1. For even length l and d + 2 ≤ l ≤ 2n−1.
By induction hypothesis, there exist n − 2 mutually inde-pendent paths {Rki}n−2i=1 of Q0n−1 with odd length k, d + 1 ≤ k ≤ 2n−1 − 1, between u and ¯v. For 1 ≤ i ≤
n − 2, we let Rk
i = u, xi,1, xi,2, . . . , xi,k−1, ¯v. Now, for
each l between d + 2 and 2n−1, we show how to
con-struct the n − 1 mutually independent paths with length l. Let P1k+1 = u, x1,1, x1,2, . . . , x1,k−1, ¯v, v, Pik+1 =
u, xi,1, xi,2, . . . , xi,k−1, ¯xi,k−1, v for 2 ≤ i ≤ n − 2, and
Pn−1k+1= u, ¯u, ¯x1,1, ¯x1,2, . . . , ¯x1,k−1, v, d+2 ≤ k+1 ≤ 2n−1.
Setl = k + 1. So, {Pil}n−1i=1 formn − 1 mutually independent
paths with each even lengthl, d + 2 ≤ l ≤ 2n−1, joining from
u to v.
Case 2. For even length l and 2n−1+ 2 ≤ l ≤ d + 2n−1− 2.
According to induction hypothesis, there existn − 2 mutually independent paths {Ri}n−2i=1 ofQ0n−1 with odd lengthd − 1
betweenu and ¯v. Without loss of generality, we write Ri =
u, xi,1, xi,2, . . . , xi,d−2, ¯v for 1 ≤ i ≤ n − 2. For each m,
2 ≤ m ≤ d − 2 and m is even, by Lemma 3, there exist n − 2 mutually independent hamiltonian paths {Si}n−2i=1 of Q1n−1
beginning at v such that {Si : v → ¯xi,m}n−2i=1. For 1 ≤ i ≤
n − 2, let Pim+2n−1 = u, xi,1, xi,2. . . , xi,m, ¯xi,m, (Si)−1, v,
2n−1+ 2 ≤ m + 2n−1 ≤ d + 2n−1− 2. Set l = m + 2n−1.
We have the firstn − 2 mutually independent paths with each even lengthl, 2n−1+ 2 ≤ l ≤ d + 2n−1− 2 joining u to v.
Finally, we construct the (n − 1)th path joining u to v. Let z be any white vertex in Q1n−1. By Lemma 4, there exists a path Tk ofQ1
n−1− {v, z} with any odd length k, 1 ≤ k ≤
d − 3, joining ¯u to ¯xn−1,1, and by Theorem 3, there exists a
hamiltonian pathU of Q0n−1− {u} between xn−1,1 to¯v. Let
Pn−1k+2n−1+1= u, ¯u, Tk, ¯x
n−1,1, xn−1,1, U, ¯v, v, 2n−1+ 2 ≤
k + 2n−1+ 1 ≤ d + 2n−1− 2. Set l = k + 2n−1+ 1. So,
{Pl
i}n−1i=1 form n − 1 mutually independent paths with each
even lengthl, 2n−1+ 2 ≤ l ≤ d + 2n−1− 2, joining from u tov.
Case 3. For even length l and d + 2n−1− 2 ≤ l ≤ 2n− 2.
Again, by induction hypothesis, there exist n − 2 mu-tually independent paths {Rmi }n−2i=1 between u and ¯v in Q0n−1 with odd length m, d − 1 ≤ m ≤ 2n−1 − 1.
Let Rm
i = u, xi,1, xi,2, . . . , xi,m−1, ¯v for 1 ≤ i ≤
n − 2. By Lemma 3, there exist n − 2 mutually inde-pendent hamiltonian paths {Si}n−2i=1 of Q1n−1 beginning at v such that {Si : v → ¯xi,m−1}n−2i=1. Let Pm+2
n−1−1
i =
u, xi,1, xi,2, . . . , xi,m−1, ¯xi,m−1, (Si)−1, v for 1 ≤ i ≤ n −
2, d+2n−1−2 ≤ m+2n−1−1 ≤ 2n−2. Set l = m+2n−1−1.
We have the firstn − 2 mutually independent paths with each even lengthl, d+2n−1−2 ≤ l ≤ 2n−2 joining u to v. Finally,
we construct the(n−1)th path joining u to v. Assume that z is any white vertex inQ1n−1. According to Lemma 4, there exists
a pathTk ofQ1
n−1− {v, z} with any odd length k, d − 3 ≤
k ≤ 2n−1− 3, joining ¯u to ¯x
n−1,1, and by Theorem 3, there
exists a hamiltonian pathU of Q0n−1− {u} between xn−1,1
to ¯v. Let Pn−1k+2n−1+1 = u, ¯u, Tk, ¯x
n−1,1, xn−1,1, U, ¯v, v,
d + 2n−1− 2 ≤ k + 2n−1+ 1 ≤ 2n− 2. Set l = k + 2n−1+ 1.
So,{Pl
i}n−1i=1 formn−1 mutually independent paths with each
even lengthl, d + 2n−1− 2 ≤ l ≤ 2n− 2, joining from u to
v.
Lemma 8. Let u and v be two nonadjacent vertices from different partite sets of Qn for n ≥ 4. There exist n − 1
mutually independent paths {Pl
i}n−1i=1 of Qn with any odd
length l, dQn(u, v) + 2 ≤ l ≤ 2n − 1, joining from u to v.
By Theorem 4, Lemma 5, Lemma 6, Lemma 7, and Lemma 8, we have the following theorem.
Theorem 5. Let u and v be any pair of vertices of Qn. For
dQn(u, v) ≥ n − 1, Qn is (n − 1, l)-mutually independent
bipanconnected for every l, dQn(u, v) ≤ l ≤ 2n− 1 with (l − dQn(u, v)) being even. As for dQn(u, v) ≤ n−2, it is also (n− 1, l)-mutually independent bipanconnected if l ≥ dQn(u, v) + 2, and is only (l, l)-mutually independent bipanconnected if l = dQn(u, v).
IV. CONCLUSION
In this paper, we explore yet another strong property of the hypercubes. We prove that every pair of vertices u and v in the n-dimensional hypercube, with dQn(u, v) ≥ n − 1, is (n − 1, l)-mutually independent bipanconnected for every l, dQn(u, v) ≤ l ≤ |V (Qn) − 1| with (l − dQn(u, v)) being even. As fordQn(u, v) ≤ n − 2, it is also (n − 1, l)-mutually independent bipanconnected ifl ≥ dQn(u, v) + 2, and is only (l, l)-mutually independent bipanconnected if l = dQn(u, v). Our result strengthens a previous results of Sun et al. [14], and Li et al. [9]. Li et al. [9] proved that the hypercubeQnis
bipanconnected forn ≥ 2. Sun et al. [14] proved that there are n − 1 mutually independent hamiltonian paths in Qnbetween
every two vertices from different partite sets forn ≥ 4. The number “n − 1” in our result is tight as we have the following observation. Because each vertex of the hypercube Qn has
exactlyn edges incident with it, we can expect at most n − 1
mutually independent paths when the given two vertices are adjacent.
ACKNOWLEDGMENT
This research was partially supported by the National Sci-ence Council of the Republic of China under contract NSC 96-2221-E-009-137-MY3, and the Aiming for the Top University and Elite Research Center Development Plan.
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