The Mutually Independent Edge-Bipancyclic
Property in Hypercube Graphs
Yuan-Kang Shih, Lun-Min Shih,
and Jimmy J. M. Tan
Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan 30010, R.O.C.
Email:{ykshih, lmshih, jmtan}@cs.nctu.edu.tw
Lih-Hsing Hsu
Department of Computer Science and Information Engineering,
Providence University, Taichung, Taiwan 43301, R.O.C.
Email: lhhsu@cs.pu.edu.tw
Abstract—A graph G is edge-pancyclic if each edge
lies on cycles of all lengths. A bipartite graph is
edge-bipancyclic if each edge lies on cycles of every even length
from 4 to |V (G)|. Two cycles with the same length m,
C1 = hu1, u2, · · · , um, u1i and C2 = hv1, v2, · · · , vm, v1i
passing through an edge (x, y) are independent with respect
to the edge (x, y) if u1= v1= x, um= vm= y and ui6= vi
for 2 ≤ i ≤ m−1. Cycles with equal length C1, C2, · · · , Cn
passing through an edge (x, y) are mutually independent
with respect to the edge (x, y) if each pair of them are
independent with respect to the edge (x, y). We propose a new concept called mutually independent edge-bipancyclicity. We say that a bipartite graph G is k-mutually independent
edge-bipancyclic if for each edge (x, y) ∈ E(G) and for
each even length l, 4 ≤ l ≤ |V (G)|, there are k cycles with the same length l passing through edge (x, y), and these k cycles are mutually independent with respect to the edge (x, y). In this paper, we prove that the hypercube Qn is
(n − 1)-mutually independent edge-bipancyclic for n ≥ 4.
Index Terms—hypercube, bipancyclic, edge-bipancyclic,
mutually independent.
I. INTRODUCTION
For the graph definitions and notations we refer the reader to [1]. A graph is denoted by G with the vertex set V (G) and the edge set E(G). The simu-lation 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 ring embedding deals with finding all the possible length of cycles in an interconnection network [2]–[4].
A path P = hv0, v1, · · · , vmi is a se-quence of adjacent vertices. We also write P =
hv0, · · · , vi, Q, vj, · · · , vmi where Q is a path
hvi, · · · , vji. A cycle C = hv0, v1, · · · , vm, v0i is a
sequence of adjacent vertices. The length of a path
P is the number of edges in P . The length of a
cycle C is the number of edges in C.
A path is a hamitonian path if it contains all the vertices of G. 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 graph G is hamiltonian laceable if there exists a hamiltonian path joining any two vertices from different partite sets. A graph G is pancyclic [1] if G includes cycles of all lengths. A graph G is called edge-pancyclic if each edge lies on cycles of all lengths. If these cycles are restricted to even length, G is called a bipancyclic graph. A bipartite graph is edge-bipancyclic [5] if each edge lies on cy-cles of every even length from 4 to |V (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 [6]. 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 and y, there exists a path of length l joining x and y, for every l,
dG(x, y) ≤ l ≤ |V (G)| − 1 and (l − dG(x, y)) being even. It is proved that the hypercube is
bipanconnected [7].
We now introduce a relatively new concept. Two paths P1 = hu1, u2, · · · , umi and P2 =
hv1, v2, · · · , vmi from a to b are independent [8] if u1 = v1 = a, um = vm = b, and ui 6= vi for 2 ≤ i ≤ m − 1. Paths with equal length
P1, P2, · · · , Pnfrom a to b are mutually independent [8] if every two different paths are independent. Two paths P1 and P2 are fully independent [9] if
ui 6= vi for all 1 ≤ i ≤ m. Paths with equal length P1, P2, · · · , Pn, are mutually fully
indepen-dent if each pair of them are fully indepenindepen-dent.
Two cycles C1 = hu1, u2, · · · , um, u1i and C2 =
hv1, v2, · · · , vm, v1i passing through an edge (x, y)
are independent with respect to the edge (x, y), if u1 = v1 = x, um = vm = y and ui 6= vi for 2 ≤ i ≤ m − 1. Cycles with equal length
C1, C2, · · · , Cn passing through an edge (x, y) are
mutually independent with respect to the edge (x, y)
if every two cycles are independent with respect to the edge (x, y).
An n-dimensional hypercube, denoted by Qn, is a graph with 2n vertices, and each vertex u 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 in ith dimension and the edge (u, v) is called an ith dimension edge. We use Q0
n to denote the subgraph of Qn induce by {u ∈ V (Qn) | ui = 0} and Q1n the subgraph of Qn induced by {u ∈ V (Qn) | ui = 1}.
Q0
n and Q1n are all isomorphic to Qn−1. Qn can be decomposed into Q0
n and Q1n by dimension i, and
Q0
n and Q1n are (n − 1)-dimensional subcubes of
Qn induced by the vertices with the ith bit position being 0 and 1 respectively. For each vertex u in
Qi
n, i = 0, 1, there is exactly one vertex in Qi−1n , denoted by ¯u, such that (u, ¯u) is an edge in Qn. Saad and Schultz [10] proved Qn is edge-bipancyclic in the sense that each edge lies on cycles of every even length from 4 to 2n. Li et al. [7] considered an injured n-dimensional hypercube Qnwhere each edge lies on cycles of every even length from 4 to 2n with n−2 edge faults. Tsai [11] proved that such injured hypercube Qncontains a cycle of every even length from 4 to 2n, even if it has up to (2n − 5)
edge faults with some specified conditions. Sun et al. [12] proved that the n-dimensional hypercube
Qncontains n−1 mutually independent hamiltonian paths between any vertex pair {x, y}, where x and
y belong to different partite sets and n ≥ 4. Let |F |
be the number of the faulty edges. Hsieh and Weng [13] showed that when 1 ≤ |F | ≤ n−2, there exists
n − |F | − 1 mutually independent hamiltonian paths
joining x to y in Qn− F , where x and y belong to different partite sets.
We now introduce a new concept. We say that a bipartite graph G is n-mutually independent
edge-bipancyclic if for each edge (x, y) ∈ E(G), and for
each even length l, 4 ≤ l ≤ |V (G)|, there are n cycles with the same length l passing through edge (x, y), and these n cycles are mutually independent with respect to the edge (x, y). In this paper, we show that the hypercube has a stronger property of edge-bipancyclic property. We prove that an n-dimensional hypercube Qn, for n ≥ 4, is (n − 1)-mutually independent edge-bipancyclic in the sense that each edge of Qn lies on n − 1 mutually independent cycles of every even length from 4 to 2n. Our result strengthens a previous result of Saad and Schultz [10]. Because each vertex of the hypercube Qn has exactly n edges incident with it, we can expect at most n − 1 mutually independent cycles passing through edge (x, y). Therefore, the result “n − 1” is tight.
II. PRELIMINARIES
In order to prove our claim, we need the following previous results.
Lemma 1. [14] The hypercube Qn is hamiltonian
laceable for every positive integer n.
Lemma 2. [7] The hypercube Qn is
bipancon-nected for n ≥ 2.
The hypercube Qn is known to be a bipartite graph. Let (B, W ) be the vertex bipartition of Qn. Edges e1, e2, · · · , en in a graph G are called
inde-pendent edges if these edges are pairwise disjoint.
Lemma 3. [12] Let {ei | 1 ≤ i ≤ n − 1} be any
n − 1 independent edges of Qn with n ≥ 2 and
ei = (bi, wi). Then there exist n − 1 mutually fully
independent hamiltonian paths Pl
1, · · · , Pn−1l of Qn
such that Pl
Theorem 1. [12] 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. [15] Let Fv be a set of faulty vertices
in Qn. There exists a path of every odd length from 3 to 2n− 2|F
v| − 1 joining any two adjacent
fault-free vertices in Qn− Fv even if |Fv| ≤ n − 2, where
n ≥ 3.
Lemma 4. [12] Qn− {x, y} is hamiltonian
lace-able, if x and y are any two vertices from different partite sets of Qn with n ≥ 4.
III. MUTUALLY INDEPENDENT
EDGE-BIPANCYCLIC PROPERTY OF HYPERCUBES
To prove our main result, we need the following lemmas, Lemma 5 to 7.
Lemma 5. Let x and y be two vertices from different partite sets of Qn with 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}.
By Theorem 2 and Lemma 4, we can prove Lemma 5 easily.
Lemma 6. Let e1 and e2 be two independent edges
of Q3, ei = (bi, wi) for i = 1, 2. Then Q3 contains
2 mutually fully independent paths Pl
1 and P2l with
any odd length l ≤ 23 such that Pl
i joins from bi to
wi, i = 1, 2.
Lemma 7. Let {ei | 1 ≤ i ≤ n − 1} be n − 1
independent edges of Qn with n ≥ 2, ei = (bi, wi),
i = 1 to n − 1. Then there exist n − 1 mutually fully independent paths Pl
1, · · · , Pn−1l of Qnwith any odd
length l ≤ 2n− 1 such that Pl
i joins from bi to wi,
i = 1 to n − 1.
Proof: It is clear that the result holds for Q2.
We prove the statement by induction on n. By Lemma 6, the statement holds for n = 3. Suppose that the result holds for Qn−1, for some n ≥ 4. The hypercube Qn has n dimensions, and there are only
n − 1 independent edges, so there is at least one
dimension which does not contain any one of these
n − 1 independent edges. We can choose one of
these dimensions to separate Qn into two (n − 1)-dimensional subcubes Q0
n and Q1n. We then prove the result by considering the following three cases.
Case 1. For odd length l and 1 ≤ l ≤ 2n−1− 1. Case 1.1. Suppose that there are k independent edges in Q0
n with 1 ≤ k ≤ n − 2 and there are
n − k − 1 independent edges in Q1
n. By induction hypothesis, the case is obvious.
Case 1.2. Without loss of generality, suppose that all the n−1 independent edges are in Q0
n. By induction hypothesis, there exist n−2 mutually fully indepen-dent paths Pl
1, · · · , Pn−2l of Q0n with any odd length
l ≤ 2n−1− 1 such that Pl
i joins from bi to wi for 1 ≤ i ≤ n − 2. By Lemma 2, there is a path Rm of
Q1
n with odd length 1 ≤ m ≤ 2n−1− 3 joining ¯bn−1 to ¯wn−1. Let Pn−1l = hbn−1, ¯bn−1, Rm, ¯wn−1, wn−1i, then 3 ≤ l ≤ 2n−1− 1. Note that, b
n−1 and wn−1 are adjacent vertices, so we obtain paths Pl
n−1 for all odd lengths l, 1 ≤ l ≤ 2n−1 − 1. Therefore, there are n − 1 mutually fully independent paths
Pl
1, · · · , Pn−1l of Qn with any odd length l ≤ 2n− 1 such that Pl
i joins from bi to wi, i = 1 to n − 1. Case 2. For odd length l and 2n−1+ 1 ≤ l ≤ 2n− 3. Case 2.1. Suppose that there are k independent edges in Q0
n with 1 ≤ k ≤ n − 2 and there are
n − k − 1 independent edges in Q1
n. By induction hypothesis, there exist k mutually fully independent paths R1, · · · , Rk of Q0n with length 2n−1− 1 such that Ri joins from bi to wi for 1 ≤ i ≤ k. We let Ri = hbi, ui, vi, Zi, wii for 1 ≤ i ≤ k. According to induction hypothesis, there exist k mutually fully independent paths Tl0
1, · · · , Tl
0
k of
Q1
n with any odd length l0 ≤ 2n−1 − 3 such that Tl0
i joins from ¯ui to ¯vi for 1 ≤ i ≤ k. Therefore, Pl
i = hbi, ui, ¯ui, Tl
0
i , ¯vi, vi, Zi, wii with 2n−1 + 1 ≤ l ≤ 2n − 3 for 1 ≤ i ≤ k. Again by induction hypothesis, there exist n − k − 1 mutually fully independent paths Rk+1, · · · , Rn−1 of Q1
n with length 2n−1 − 1 such that Ri joins from bi to wi for k + 1 ≤ i ≤ n − 1. We let
Ri = hbi, ui, vi, Zi, wii for k + 1 ≤ i ≤ n − 1. By induction hypothesis, there exist n − k − 1 mutually fully independent paths Tl0
k+1, · · · , Tl
0
n−1 of
Q0
n with any odd length l0 ≤ 2n−1 − 3 such that
Tl0
i joins from ¯ui to ¯vi for k + 1 ≤ i ≤ n − 1. Therefore, Pl
i = hbi, ui, ¯ui, Tl
0
i , ¯vi, vi, Zi, wii with 2n−1 + 1 ≤ l ≤ 2n − 3 for k + 1 ≤ i ≤ n − 1. Hence, there are n − 1 mutually fully independent paths Pl
1, · · · , Pn−1l of Qn with any odd length 2n−1 + 1 ≤ l ≤ 2n − 3 such that Pl
i joins from
Case 2.2. Without loss of generality, suppose that all the n − 1 independent edges are in Q0
n. By induction hypothesis, there exist n − 2 mutually fully independent paths R1, · · · , Rn−2 of Q0n with length 2n−1− 1 such that R
i joins from bi to wi for 1 ≤ i ≤ n − 2. We let Ri = hbi, Zi, ui, vi, zi, wii. Again by induction hypothesis, there exist n − 2 mutually fully independent paths Tl0
1 , · · · , Tl
0
n−2 of
Q1
n with any odd length l0 ≤ 2n−1− 3 such that Tl
0
i joins from ¯ui to ¯vi for 1 ≤ i ≤ n − 2. Therefore,
Pl
i = hbi, Zi, ui, ¯ui, Tl
0
i , ¯vi, vi, zi, wii with any odd length 2n−1+ 1 ≤ l ≤ 2n− 3 for 1 ≤ i ≤ n − 2. By Lemma 2, there exists a path Rn−1 of Q1n with length 2n−1 − 3 joining ¯b
n−1 to ¯wn−1. We let Rn−1 = h¯bn−1, Zn−1, un−1, vn−1, ¯wn−1i. By Lemma 5, there exists a path Tl0
n−1 with every odd length 1 ≤ l0 ≤ 2n−1 − 3 joining ¯u
n−1 to ¯ vn−1 in Q0n − {bn−1, wn−1}. Therefore, Pn−1l = hbn−1, ¯bn−1, Zn−1, un−1, ¯un−1, Tl 0 n−1, ¯vn−1, vn−1, ¯wn−1, wn−1i with 2n−1+ 1 ≤ l ≤ 2n − 3. So, there are n − 1
mutually fully independent paths Pl
1, · · · , Pn−1l of
Qn with any odd length 2n−1 + 1 ≤ l ≤ 2n − 3 such that Pl
i joins from bi to wi.
Case 3. For odd length l and l = 2n− 1. This case is proved by Lemma 3.
By Case 1 Case 2 and Case 3, the proof is complete.
We now prove our main result by induction. Lemma 8. The hypercube Q4 is 3-mutually
inde-pendent edge-bipancyclic.
Theorem 3. The hypercube Qn is (n − 1)-mutually
independent edge-bipancyclic for n ≥ 4.
Proof: Let (u, v) be an edge in Qn, n ≥ 4. We prove the statement by induction on n. By Lemma 8, the statement holds for n = 4. Suppose that the result holds for Qn−1, n ≥ 5. We may choose a dimension to divide the hypercube Qn into two subcubes Q0
nand Q1nso that the edge (u, v) is in Q0n. According to the length l of the cycles, we divide the proof into the following three cases. In each case, the length l is assumed to be an even number. We shall find n − 1 mutually independent cycles with length l passing through edge (u, v).
Case 1. For even length l and 4 ≤ l ≤ 2n−1. By induction hypothesis, there exist n − 2 mutually independent cycles with respect to the edge (u, v),
Ck
1, · · · , Cn−2l with any even length 4 ≤ l ≤ 2n−1 in
Q0
n. By Lemma 2, there is a path Pkof Q1nwith any odd length 1 ≤ k ≤ 2n−1−3 joining ¯u to ¯v. Then we have Cl
n−1= hu, ¯u, Pk, ¯v, v, ui with any even length 4 ≤ l ≤ 2n−1. Therefore, there exist n − 1 mutually independent cycles with respect to the edge (u, v),
Cl
1, · · · , Cn−1l with every even length 4 ≤ l ≤ 2n−1. Case 2. For even length l and 2n−1+2 ≤ l ≤ 2n−2. By induction hypothesis, there exist n − 2 mutu-ally independent cycles with respect to the edge (u, v), R1, · · · , Rn−2 with length 2n−1 of Q0n. We let Ri = hu, xi, yi, zi, · · · , v, ui for 1 ≤ i ≤ n − 2. By Lemma 7, for any given odd length k ≤ 2n−1−3 there exist n − 2 mutually fully independent paths
Pk
1, · · · , Pn−2k all with the same length k, such that
Pk
i joins from ¯yi to ¯zi for 1 ≤ i ≤ n − 2. We let Cl
i = hui, xi, yi, ¯yi, Pik, ¯zi, zi, Ri, vi, uii. Then Cil,
i = 1 to n − 2 are with any even length l, where
2n−1 + 2 ≤ l ≤ 2n − 2. By Lemma 1, there exists a hamiltonian path P0 of Q1
n joining ¯u to ¯v. Let P0 = h¯u, y
n−1, zn−1, T, ¯vi. By Lemma 5, there exists a path Uk0
with every odd length 1 ≤ k0 ≤ 2n−1− 3 joining ¯y n−1 to ¯zn−1 in Q0n− {u, v}. We let Cl n−1= hu, ¯u, yn−1, ¯yn−1, Uk 0 , ¯zn+1, zn+1, T, ¯v, vi with any even length l, 2n−1+2 ≤ l ≤ 2n−2. Hence, there exist n − 1 mutually independent cycles with respect to edge (u, v), Cl
1, · · · , Cn−1l with any even length 4 ≤ l ≤ 2n−1.
Case 3. For even length l and l = 2n. This case is proved by Theorem 1.
By Case 1, Case 2 and Case 3, we complete the proof.
IV. CONCLUSION
In [1], the author introduced a popular property called the pancyclicity. A stronger property is
edge-bipancyclicity which was proposed by Mitchem and
Schmeichel in [5]. Another interesting property is the mutually independent paths. Sun et al. [12] proved that the n-dimensional hypercube graph contains n − 1 mutually independent hamiltonian paths between any vertex pair {x, y}, where x and
y belong to different partite sets and n ≥ 4. In
this paper, we combine the two properties, edge-bipancyclicity and mutually independent paths, into a new stronger property called mutually
hypercube Qnis (n−1)-mutually independent edge-pancyclic for n ≥ 4. Our result also strengthens a previous result of Saad and Schultz [10], in the sense that the hypercube Qn is not only bipancyclic but also mutually independent edge-pancyclic.
V. Acknowledgements
This research was partially supported by the National Science Council of the Republic of China under contract NSC 96-2221-E-009-137-MY3, and the Aiming for the Top University and Elite Re-search Center Development Plan.
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