www.elsevier.com/locate/ipl
Node-pancyclicity and edge-pancyclicity of hypercube variants
Ken S. Hu
a, Shyun-Shyun Yeoh
a, Chiuyuan Chen
a,∗,1, Lih-Hsing Hsu
baDepartment of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan
bDepartment of Computer Science and Information Engineering, Providence University, Taichung County 433, Taiwan
Received 21 January 2006; received in revised form 22 October 2006; accepted 24 October 2006 Available online 27 November 2006
Communicated by A.A. Bertossi
Abstract
Twisted cubes, crossed cubes, Möbius cubes, and locally twisted cubes are some of the widely studied hypercube variants. The 4-pancyclicity of twisted cubes, crossed cubes, Möbius cubes, locally twisted cubes and the 4-edge-pancyclicity of twisted cubes, crossed cubes, Möbius cubes are proven in [C.P. Chang, J.N. Wang, L.H. Hsu, Topological properties of twisted cube, Inform. Sci. 113 (1999) 147–167; C.P. Chang, T.Y. Sung, L.H. Hsu, Edge congestion and topological properties of crossed cubes, IEEE Trans. Parall. Distr. 11 (1) (2000) 64–80; J. Fan, Hamilton-connectivity and cycle embedding of the Möbius cubes, Inform. Process. Lett. 82 (2002) 113–117; X. Yang, G.M. Megson, D.J. Evans, Locally twisted cubes are 4-pancyclic, Appl. Math. Lett. 17 (2004) 919–925; J. Fan, N. Yu, X. Jia, X. Lin, Embedding of cycles in twisted cubes with edge-pancyclic, Algorithmica, submitted for publication; J. Fan, X. Lin, X. Jia, Node-pancyclic and edge-pancyclic of crossed cubes, Inform. Process. Lett. 93 (2005) 133–138; M. Xu, J.M. Xu, Edge-pancyclicity of Möbius cubes, Inform. Process. Lett. 96 (2005) 136–140], respectively. It should be noted that 4-edge-pancyclicity implies 4-node-pancyclicity which further implies 4-pancyclicity. In this paper, we outline an approach to prove the 4-edge-pancyclicity of some hypercube variants and we prove in particular that Möbius cubes and locally twisted cubes are 4-edge-pancyclic.
©2006 Elsevier B.V. All rights reserved.
Keywords: Interconnection networks; Hypercube; Crossed cube; Möbius cube; Locally twisted cube; Pancyclicity
1. Introduction
Interconnection networks are essential for parallel and distributed computing. The hypercube is one of the most popular interconnection networks since it has sim-ple structure and is easy to imsim-plement. An interconnec-tion network can be represented by a graph G= (V, E),
* Corresponding author.
E-mail address: cychen@mail.nctu.edu.tw (C. Chen).
1 This research was partially supported by the National Science
Council of the Republic of China under the grant NSC94-2115-M-009-006.
where V is the set of nodes and E is the set of edges of the network. In this paper, we will use graphs and inter-connection networks interchangeably.
It has been shown that hypercubes do not achieve the smallest possible diameter for its resources. Therefore, many variants were proposed. The most well-known variants are twisted cubes [9], crossed cubes [4], and Möbius cubes [3]; they have diameters about half of that of a hypercube. Generally, the drawback of these vari-ants is that the labels of some neighboring nodes may differ in as many as n/2 bits, where n is the dimen-sion of these hypercube variants (see [11] for details). For example, in the 10-dimensional crossed cube, nodes
0020-0190/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2006.10.008
0001010101 and 1011111111 are adjacent and they dif-fer in 5 bits. Based on this observation, Yang et al. [11] proposed the locally twisted cubes with diameters about half of that of a hypercube, of which the labels of any two neighboring nodes differ in at most two successive bits.
The following terminologies will be used through-out this paper. An -cycle is a cycle of length . Let
G= (V, E) be a graph and L |V | be a positive
in-teger. G is L-pancyclic if for every integer ∈ {L,
L+ 1, . . . , |V |}, G contains an -cycle. G is L-node-pancyclic if for every node x ∈ V and every integer ∈ {L, L + 1, . . . , |V |}, G contains an -cycle C such
that x is in C. G is L-edge-pancyclic if for every edge
(x, y)∈ E and every integer ∈ {L, L + 1, . . . , |V |}, G
contains an -cycle C such that (x, y) is in C.
One way to evaluate an interconnection network (a host graph) is to see how well other existing networks (the guest graphs) can be embedded into it. Graph em-bedding can be formally defined as follows: Given two graphs G= (V, E) and H = (V, E), an embedding from G to H is a mapping ψ : V → V. An important benefit of graph embedding is that we can apply existing algorithms for the guest graphs to the host graph. Cycles (i.e., rings) and trees are commonly used guest graphs. This paper will discuss the cycle-embedding properties of Möbius cubes and locally twisted cubes (these cubes will be defined later).
Twisted cubes, crossed cubes, Möbius cubes, and lo-cally twisted cubes are superior to hypercubes when the cycle-embedding capability is considered. The 4-pan-cyclicity of twisted cubes, crossed cubes, Möbius cubes, and locally twisted cubes are proven in [2,1,5,12], re-spectively. Recently, Fan et al. [6] proved that crossed cubes are not only 4-node-pancyclic but also 4-edge-pancyclic. It should be noted that 4-edge-pancyclicity implies 4-node-pancyclicity (thus the proof in [6] for the 4-node-pancyclicity of crossed cubes is actually re-dundant) which further implies 4-pancyclic. Lately, the 4-edge-pancyclicity of twisted cubes and Möbius cubes are proven in [7,10], respectively (see also [8]).
In this paper, we outline an approach to prove the 4-edge-pancyclicity of some hypercube variants and we prove, in particular, that Möbius cubes and locally twisted cubes are 4-edge-pancyclic. We also show how to use our approach to prove that crossed cubes are 4-edge-pancyclic.
This paper is organized as follows. In Section 2, we give some definitions and notations. In Section 3, we outline an approach to prove 4-edge-pancyclicity. In Sections 4–6, we prove that locally twisted cubes,
crossed cubes, and Möbius cubes are 4-edge-pancyclic. The final section concludes this paper.
2. Preliminaries
Let G= (V, E) be a graph and let L |V | − 1 be a positive integer. G is L-path-connected if G contains a path of length L between any two distinct nodes.
G is Hamiltonian-connected if G is (|V | −
1)-path-connected.
The n-dimensional hypercube Qnis a graph with 2n
nodes and n· (2n−1)edges such that its nodes are n-tu-ples with entries in{0, 1} and its edges are the pairs of
n-tuples that differ in exactly one position. Thus Q1is
the complete graph with two nodes 0 and 1, and Qn
(n 2) is built from two copies of Qn−1 as follows:
Let k∈ {0, 1} and let kQn−1denote the graph obtained
by prefixing the label of each node of one copy of Qn−1
with k; connect each node 0xn−1. . . x2x1of 0Qn−1with
the node 1xn−1. . . x2x1of 1Qn−1by an edge.
We now define a generalization of Qn. The
n-di-mensional general cube GQn is defined recursively as
follows (see Fig. 1). GQ1 is Q1, and GQn (n 2) is
built from two GQn−1’s (not necessarily identical) as
follows: Let k∈ {0, 1} and let kGQn−1denote the graph
obtained by prefixing the label of each node of one of the two GQn−1’s with k; add a perfect matching
be-tween 0GQn−1and 1GQn−1, i.e., each node in 0GQn−1
is adjacent to exactly one node in 1GQn−1.
We assume conventionality of the node prefixing method kGQn−1 which will be used repeatedly in the
definitions of specific hypercube variants late in this paper unless otherwise specified. We will see in the fol-lowing sections that crossed cubes, Möbius cubes, and locally twisted cubes are the examples of GQn. Note
that the two GQn−1’s in GQnare not necessarily
identi-cal. For instance, for crossed cubes and locally twisted cubes, the two GQn−1’s are identical; but for Möbius
cubes, they are not.
For clarity, let V (G) and E(G) denote the set of nodes and the set of edges of G, respectively. We
say that (x, y) is a matching edge in GQn if x ∈
V (0GQn−1), y∈ V (1GQn−1), and x is matched with
y. If (x, y) is a matching edge, then we write m(x) for
y and m(y) for x. We say that GQn has the 4-cycle
property if for every matching edge (x, y), there
ex-ists a matching edge (u, v) such that (x, u, v, y, x) form a 4-cycle in GQn. We say that GQn has the 5-cycle
property if for every matching edge (x, y), there exist
a matching edge (s, t) and a node r∈ V (0GQn−1)such
that (x, r, s, t, y, x) form a 5-cycle in GQn.
3. 4-edge-pancyclicity of general cubes
In this section, we outline an approach to prove 4-edge-pancyclicity. We first give two lemmas.
Lemma 1. For n 4, if both 0GQn−1 and 1GQn−1
are Hamiltonian-connected, then GQn is
Hamiltonian-connected.
Proof. Let x and y be two arbitrary distinct nodes of
GQn. Then there are four cases.
Case 1. x∈ V (0GQn−1)and y∈ V (0GQn−1). Since
0GQn−1 is Hamiltonian-connected, it has a
Hamil-tonian path (p1, p2, . . . , p2n−1) such that p1= x and
p2n−1 = y. Since 1GQn−1 is Hamiltonian-connected,
it has a Hamiltonian path (q1, q2, . . . , q2n−1)such that
q1= m(p1)and q2n−1 = m(p2). Hence (x, q1, q2, . . . ,
q2n−1, p2, p3, . . . , p2n−1−1, y)is a Hamiltonian path be-tween x and y in GQn.
Case 2. x∈ V (1GQn−1)and y∈ V (1GQn−1). The
argument is similar to that of Case 1.
Case 3. x ∈ V (0GQn−1) and y ∈ V (1GQn−1).
Let z∈ V (0GQn−1) such that z= x. Since 0GQn−1
is Hamiltonian-connected, it has a Hamiltonian path
(p1, p2, . . . , p2n−1) such that p1 = x and p2n−1 = z.
Since 1GQn−1 is Hamiltonian-connected, it has a
Ha-miltonian path (q1, q2, . . . , q2n−1)such that q1= m(z)
and q2n−1 = y. Hence (x, p2, . . . , p2n−1, q1, q2. . . ,
q2n−1−1, y)is a Hamiltonian path between x and y in
GQn.
Case 4. x∈ V (1GQn−1)and y∈ V (0GQn−1). The
argument is similar to that of Case 3. 2
Lemma 2. For n 4, if both 0GQn−1and 1GQn−1are
Hamiltonian-connected and (2n−1−2)-path-connected,
then GQnis (2n− 2)-path-connected.
Proof. Let x and y be two arbitrary distinct nodes of
GQn. Then there are four cases.
Case 1. x∈ V (0GQn−1)and y∈ V (0GQn−1). Since
0GQn−1 is Hamiltonian-connected, it has a
Hamil-tonian path (p1, p2, . . . , p2n−1) such that p1= x and
p2n−1= y. Since 1GQn−1is (2n−1−2)-path-connected,
it has a path (q1, q2, . . . , q2n−1−1) of length 2n−1− 2 such that q1 = m(p1) and q2n−1−1 = m(p2). Hence
(x, q1, q2, . . . , q2n−1−1, p2, p3, . . . , p2n−1−1, y)is a path of length 2n− 2 between x and y in GQn.
Case 2. x∈ V (1GQn−1)and y∈ V (1GQn−1). The
argument is similar to that of Case 1.
Case 3. x ∈ V (0GQn−1) and y ∈ V (1GQn−1).
Let z∈ V (0GQn−1) such that z= x. Since 0GQn−1
is Hamiltonian-connected, it has a Hamiltonian path
(p1, p2, . . . , p2n−1) such that p1= x and p2n−1 = z.
Since 1GQn−1 is (2n−1− 2)-path-connected, it has a
path (q1, q2, . . . , q2n−1−1)of length 2n−1− 2 such that q1= m(z) and q2n−1−1= y. Hence (x, p2, . . . , p2n−1, q1, q2. . . , q2n−1−2, y)is a path of length 2n− 2 between xand y in GQn.
Case 4. x∈ V (1GQn−1)and y∈ V (0GQn−1). The
argument is similar to that of Case 3. 2
We now outline an approach to prove the 4-edge-pancyclicity of GQn.
Theorem 3. For n 4, if all the GQ3’s in GQn are
4-edge-pancyclic, Hamiltonian-connected, and (23−
2)-path-connected, and if GQn has both the 4-cycle and
the 5-cycle properties, then GQnis 4-edge-pancyclic.
Proof. This theorem follows from Lemmas 1, 2, and
the following claim.
Claim. For n 4, if both 0GQn−1 and 1GQn−1 are
4-edge-pancyclic, Hamiltonian-connected, and (2n−1− 2)-path-connected, and if GQn has both the 4-cycle
property and the 5-cycle property, then GQnis
4-edge-pancyclic.
We now prove the claim. Let (x, y) be an arbitrary edge of E(GQn) and let ∈ {4, 5, . . . , 2n}. There are
four cases.
Case 1. x∈ V (0GQn−1)and y∈ V (0GQn−1). Then
there are three subcases.
Subcase 1.1. 4 2n−1. Since 0GQn−1is
4-edge-pancyclic, there exists an -cycle that contains (x, y) in 0GQn−1, hence in GQn.
Subcase 1.2. = 2n−1+ 1. Let u = m(x) and v = m(y). Since 1GQn−1 is (2n−1− 2)-path-connected, it
has a path (p1, p2, . . . , p2n−1−1)of length 2n−1−2 such
that p1= v and p2n−1−1= u. Thus (x, y, p1, p2, . . . ,
p2n−1−1, x)is a (2n−1+ 1)-cycle in GQn that contains
Subcase 1.3. 2n−1 + 2 2n. Since 0GQ n−1
is 4-edge-pancyclic and (x, y) is an edge in 0GQn−1,
there exists a 2n−1-cycle C= (p1, p2, . . . , p2n−1, p1)in
0GQn−1 such that p1= x and p2= y. Note that 1
− 2n−1− 1 2n−1− 1. Let (p1, p2, . . . , p−2n−1)be
the path of length −2n−1−1 in C. Set w = p
−2n−1for easy writing. Let u= m(x) and v = m(w). Then u, v ∈
V (1GQn−1). Since 1GQn−1is Hamiltonian-connected,
there is a path (q1, q2, . . . , q2n−1) of length 2n−1− 1
in 1GQn−1 such that q1 = v and q2n−1 = u. Thus (p1, p2, . . . , p−2n−1, q1, q2, . . . , q2n−1, p1)is a cycle of
length (− 2n−1− 1) + 1 + (2n−1− 1) + 1 = in GQn
that contains (x, y).
Case 2. x∈ V (1GQn−1)and y ∈ V (1GQn−1). The
argument is similar to that of Case 1.
Case 3. x∈ V (0GQn−1)and y∈ V (1GQn−1). Then
there are four subcases.
Subcase 3.1. ∈ {4, 5}. Since GQn has the 4-cycle
property and the 5-cycle property, there exists a cycle of length in GQnthat contains (x, y).
Subcase 3.2. 6 2n−1+ 2. Since GQn has
the 4-cycle property, there exist u∈ V (0GQn−1) and
v∈ V (1GQn−1)such that (x, u, v, y, x) form a 4-cycle
in GQn. Let m= − 2. Then 4 m 2n−1. Since
0GQn−1 is 4-edge-pancyclic, there exists a m-cycle
(p1, p2, . . . , pm, p1)in 0GQn−1such that p1= x and
pm= u. Thus (x, p2, . . . , pm, v, y, x) is an (m+
2)-cycle (i.e., an -2)-cycle) in GQnthat contains (x, y).
Subcase 3.3. = 2n−1+ 3. Since GQn has the
4-cycle property, there exist u∈ V (0GQn−1) and v ∈
V (1GQn−1) such that (x, u, v, y, x) form a 4-cycle
in GQn. Since 0GQn−1 is 4-edge-pancyclic, there
ex-ists a (2n−1 − 1)-cycle (p1, p2, . . . , p2n−1−1, p1) in
0GQn−1 such that p1 = x and p2n−1−1 = u. Since
1GQn−1 is 4-edge-pancyclic, there exists a 4-cycle
(q1, q2, q3, q4, q1) in 1GQn−1 such that q1 = v and
q4= y. Thus (p1, p2, . . . , p2n−1−1, q1, q2, q3, q4, p1)is
a (2n−1+ 3)-cycle in GQnthat contains (x, y).
Subcase 3.4. 2n−1+ 4 2n. Since GQnhas the
4-cycle property, there exist u∈ V (0GQn−1) and v∈
V (1GQn−1)such that (x, u, v, y, x) form a 4-cycle in
GQn. Since 0GQn−1is 4-edge-pancyclic, there exists a
2n−1-cycle (p1, p2, . . . , p2n−1, p1)in 0GQn−1such that
p1= x and p2n−1= u. Let m = −2n−1. Then 4 m
2n−1. Since 1GQ
n−1is 4-edge-pancyclic, there exists a
m-cycle (q1, q2, . . . , qm, q1)in 1GQn−1such that q1=
vand qm= y. Thus (p1, p2, . . . , p2n−1, q1, q2, . . . , qm)
is a cycle of length (2n−1− 1) + (m − 1) + 2 = m + 2n−1= in GQnthat contains (x, y).
Case 4. x∈ V (1GQn−1)and y ∈ V (0GQn−1). The
argument is similar to that of Case 3. 2
Fig. 2. (a) LTQ3. (b) A symmetric drawing of LTQ3.
Fig. 3. LTQ4.
4. Pancyclicity of locally twisted cubes
The purpose of this section is to use Theorem 3 to prove that locally twisted cubes are 4-edge-pancyclic.
The n-dimensional locally twisted cube LTQnis
de-fined recursively as follow. LTQ1is Q1, and LTQ2is the
graph consisting of four nodes labeled with 00, 01, 10, and 11, respectively, and connected by the four edges (00, 01) (00, 10), (01, 11), and (10, 11). LTQn (n 3)
is built from two identical LTQn−1’s as follows: connect
each node 0xn−1xn−2. . . x1of 0LTQn−1with the node
1(xn−1+ x1)xn−2. . . x1of 1LTQn−1by an edge, where
‘+’ means the modulo 2 addition operation. See Figs. 2 and 3 for examples.
Before going any further, we work out the adjacency relation of LTQn. For convenience, xidenotes the
com-plement of xi.
Lemma 4. For every x= xnxn−1. . . x1∈ V (LTQn), the
n nodes y1, y2, . . . , ynadjacent to x are:
y1= xnxn−1xn−2. . . x3x2x1, y2= xnxn−1xn−2. . . x3x2x1, y3= xnxn−1xn−2. . . x3(x2+ x1)x1, .. . yn−1= xnxn−1(xn−2+ x1) . . . x3x2x1, yn= xn(xn−1+ x1)xn−2. . . x3x2x1.
Proof. By the definition of LTQn, (x, yn)∈ E(LTQn).
(x, y1)∈ E(LTQn) because (x1, x1)∈ E(LTQ1) and
because (x2x1, x2x1)∈ E(LTQ2) and LTQn is built
from LTQ2. For 3 i n − 1, (x, yi)∈ E(LTQn)
because (xixi−1xi−2. . . x1, xi(xi−1+ x1)xi−2. . . x1)∈
E(LTQi)and LTQnis built from LTQi. 2
It is not difficult to see that: for each n, there is only one type of LTQn. Thus for n 4, all the LTQ3’s in
LTQn are identical. We are now ready to prove that
lo-cally twisted cubes satisfy Theorem 3.
Theorem 5. LTQ3 is 4-edge-pancyclic,
Hamiltonian-connected, and (23 − 2)-path-connected. For n 4, LTQn has both the 4-cycle property and the 5-cycle
property.
Proof. In [12], it was proven that LTQnis
Hamiltonian-connected and (2n− 2)-path-connected for n 3. Thus
LTQ3 is Hamiltonian-connected and (23 −
2)-path-connected. We now prove that LTQ3 is
4-edge-pan-cyclic. Since LTQ3is node-symmetric (see Fig. 2(b)), it
suffices to consider the edge (x, y)∈ {(000, 001), (000, 010)}. The cycles of lengths from 4 to 8 containing
(000, 001) (underlined) are listed as follows: length 4: 000, 001, 011, 010, 000;
length 5: 000, 001, 111, 101, 100, 000; length 6: 000, 001, 011, 010, 110, 100, 000; length 7: 000, 001, 011, 101, 111, 110, 100, 000; length 8: 000, 001, 111, 110, 010, 011, 101, 100, 000. The cycles of lengths from 4 to 8 containing (000, 010) (underlined) are listed as follows:
length 4: 000, 010, 110, 100, 000; length 5: 000, 010, 110, 111, 001, 000; length 6: 000, 010, 110, 111, 101, 100, 000; length 7: 000, 010, 110, 100, 101, 111, 001, 000; length 8: 000, 010, 110, 111, 001, 011, 101, 100, 000. Thus LTQ3is 4-edge-pancyclic.
We now prove that LTQn has the 4-cycle property
and the 5-cycle property. Let (x, y) be an arbitrary matching edge of LTQnand let x= 0xn−1xn−2. . . x2x1.
By the definition of LTQn, y = 1(xn−1+ x1)xn−2. . .
x2x1.
First consider the 4-cycle property. Let u= 0xn−1
xn−2. . . x2x1 and v= 1(xn−1+ x1)xn−2. . . x2x1. By
Lemma 4, {(x, u), (u, v), (v, y)} ⊆ E(LTQn). Hence
(x, u, v, y, x)is a 4-cycle in LTQn that contains (x, y).
Now consider the 5-cycle property. If x1 = 0, let
r= 0xn−1xn−2. . . x20, s= 0xn−1xn−2. . . x21, and t=
1xn−1xn−2. . . x21; otherwise, if x1= 1, let r = 0xn−1
xn−2. . . x20, s= 0xn−1xn−2. . . x20, and t= 1xn−1xn−2
. . . x20. By Lemma 4, {(x, r), (r, s), (s, t), (t, y)} ⊆
E(LTQn). Hence (x, r, s, t, y, x) is a 5-cycle in LTQn
that contains (x, y). 2
It was proven in [12] that LTQn is 4-pancyclic. We
now strengthen this result.
Theorem 6. For n 2, LTQnis 4-edge-pancyclic.
Proof. Clearly, this theorem holds when n= 2. By
The-orem 5, this theThe-orem holds when n= 3. For n 4, this theorem follows from Theorems 3 and 5. 2
The following corollary is obvious.
Corollary 7. For n 2, LTQnis 4-node-pancyclic.
5. Pancyclicity of crossed cubes
We first give the definition of crossed cubes. Two bi-nary strings x= x2x1 and y= y2y1 of length two are
said to be pair related (denoted by x∼ y) if and only if
(x, y)∈ {(00, 00), (10, 10), (01, 11), (11, 01)}. The n-dimensional crossed cube CQnis defined recursively as
follows. CQ1 is Q1, and CQ2 is the graph consisting
of four nodes labeled with 00, 01, 10 and 11, respec-tively, and connected by the four edges (00, 01), (00, 10), (01, 11), and (10, 11). CQn (n 3) is built from
two identical CQn−1’s as follows: connect each node
0xn−1. . . x2x1of 0CQn−1with the node 1yn−1. . . y2y1
of 1CQn−1by an edge if and only if
(1) xn−1= yn−1if n is even, and
(2) x2ix2i−1∼ y2iy2i−1for 1 i < n/2.
In [6], Fan et al. have proven that crossed cubes are 4-edge-pancyclic. We now show how to use Theorem 3 to obtain this result. It is not difficult to see that: for each n, there is only one type of CQn. Thus for n 4,
all the CQ3’s in CQnare identical. We are now ready to
prove that crossed cubes satisfy Theorem 3.
Theorem 8. CQ3 is 4-edge-pancyclic,
Hamiltonian-connected, and (23−1− 2)-path-connected. For n 4, CQnhas both the 4-cycle property and the 5-cycle
prop-erty.
Since the proof for each condition in this theorem can be found in [6], we omit the proof. We have the following theorem.
Proof. Clearly, this theorem holds when n= 2. By
The-orem 8, this theThe-orem holds when n= 3. For n 4, this theorem follows from Theorems 3 and 8. 2
By Theorem 9, it is obvious that for n 2, CQn is
4-node-pancyclic and 4-pancyclic.
6. Pancyclicity of Möbius cubes
In [10], Xu et al. have proven that Möbius cubes are 4-edge-pancyclic. In this section, we show how to use Theorem 3 to obtain this result.
The n-dimensional Möbius cube MQnis defined
re-cursively as follow (see Figs. 4 and 5): (1) MQ1is Q1.
(2) There are two types of MQ2: one is named 0-MQ2
and the other, 1-MQ2. 0-MQ2is the graph
consist-ing of four nodes labeled with 00, 01, 10, and 11, respectively, and connected by the four edges (00, 01), (00, 10), (01, 11), and (10, 11). 1-MQ2has the
same nodes as 0-MQ2, but connected by the four
edges (00, 01), (00, 11), (01, 10), and (10, 11). (3) For n 3, there are two types of MQn: 0-MQn
and 1-MQn. Both 0-MQnand 1-MQnare built from
Fig. 4. (a) 0-MQ3. (b) 1-MQ3.
Fig. 5. (a) 0-MQ4. (b) 1-MQ4.
0MQn−1and 1MQn−1with the MQn−1in 0MQn−1
being 0-MQn−1and the MQn−1in 1MQn−1being
1-MQn−1. In 0-MQn, each node 0xn−1xn−2. . . x1
of 0MQn−1is connected with the node 1xn−1xn−2
. . . x1 of 1MQn−1; while in 1-MQn, each node
0xn−1xn−2. . . x1of 0MQn−1is connected with the
node 1xn−1xn−2. . . x1of 1MQn−1.
Before going any further, we work out the adjacency relation of MQn.
Lemma 10. For every x= xnxn−1. . . x2x1∈ V (MQn),
the n nodes y1, y2, . . . , ynadjacent to x are as follows.
For 1 i n − 1, yi= x nxn−1. . . xi+1xixi−1. . . x1 if xi+1= 0, xnxn−1. . . xi+1xixi−1. . . x1 if xi+1= 1. For 0-MQn, yn= xnxn−1. . . x1; for 1-MQn, yn= xnxn−1. . . x1.
Proof. This lemma follows from the definition of
Möbius cubes given in [3]. 2
It is not difficult to see that: for each n, there are two types of MQn: the 0-MQn and the 1-MQn. Thus
for n 4, all the MQ3’s in MQn are either 0-MQ3 or
1-MQ3. We are now ready to prove that Möbius cubes
satisfy Theorem 3.
Theorem 11. Both the 0-MQ3 and the 1-MQ3 are
4-edge-pancyclic, Hamiltonian-connected, and (23− 2)-path-connected. For n 4, MQn has both the 4-cycle
property and the 5-cycle property.
Proof. From Figs. 2, 4, and 6, both 0-MQ3and 1-MQ3
are isomorphic to LTQ3. Thus by Theorem 5, both
0-MQ3 and 1-MQ3 are 4-edge-pancyclic,
Hamiltonian-connected, and (23− 2)-path-connected.
We now prove that MQnhas the 4-cycle property and
the 5-cycle property. Let (x, y) be an arbitrary matching
Fig. 6. (a) A symmetric drawing of 0-MQ3. (b) A symmetric drawing
edge of MQn and let x = 0xn−1xn−2. . . x2x1. By the
definition of MQn, y= 1xn−1xn−2. . . x2x1if this MQn
is 0-MQn and y= 1xn−1xn−2. . . x2x1 if this MQn is
1-MQn.
First consider the 4-cycle property. Let u= xnxn−1
. . . x2x1. If this MQnis 0-MQn, then let v= xnxn−1. . .
x2x1; otherwise, if this MQn is 1-MQn, then let v=
xnxn−1. . . x2x1. By Lemma 10,{(x, u), (u, v), (v, y)} ⊆
E(MQn). Hence (x, u, v, y, x) is a 4-cycle in MQnthat
contains (x, y).
Now consider the 5-cycle property. Let s = 0xn−1
xn−2. . . x2x1 and choose r and t according to the
fol-lowing rules:
1. If this MQn is 0-MQn and xn−1= 0, then let r =
0xn−1xn−2. . . x2x1and t= 1xn−1xn−2. . . x2x1.
2. If this MQn is 0-MQn and xn−1= 1, then let r =
0xn−1xn−2. . . x2x1and t= 1xn−1xn−2. . . x2x1.
3. If this MQn is 1-MQn and xn−1= 0, then let r =
0xn−1xn−2. . . x2x1and t= 1xn−1xn−2. . . x2x1.
4. If this MQn is 1-MQn and xn−1= 1, then let r =
0xn−1xn−2. . . x2x1and t= 1xn−1xn−2. . . x2x1.
By Lemma 10, {(x, r), (r, s), (s, t), (t, y)} ⊆ E(MQn).
Hence (x, r, s, t, y, x) is a 5-cycle in MQnthat contains
(x, y). 2
It was proven in [5] that MQnis 4-pancyclic. We now
strengthen this result (see also [10]).
Theorem 12. For n 2, MQnis 4-edge-pancyclic.
Proof. Clearly, this theorem holds when n= 2. By
The-orem 11, this theThe-orem holds when n= 3. For n 4, this theorem follows from Theorems 3 and 11. 2
The following corollary is obvious.
Corollary 13. For n 2, MQnis 4-node-pancyclic.
7. Concluding remarks
In this paper, we outline an approach to prove the edge-pancyclicity (hence node-pancyclicity and
4-Table 1
Pancyclicity of hypercube variations
cubes 4-pan 4-node-pan 4-edge-pan
twisted [2] [7,8] [7,8]
crossed [1] [6] [6]
Möbius [5] [10] [10]
loc twisted [12] this paper this paper
pancyclicity) of some hypercube variants. We prove in particular that Möbius cubes and locally twisted cubes are 4-edge-pancyclic. We now summarize known results on the pancyclicity properties of various hypercube vari-ants in Table 1 (in this table, “pan” means pancyclic and “loc twisted” means locally twisted).
References
[1] C.P. Chang, T.Y. Sung, L.H. Hsu, Edge congestion and topologi-cal properties of crossed cubes, IEEE Trans. Parall. Distr. 11 (1) (2000) 64–80.
[2] C.P. Chang, J.N. Wang, L.H. Hsu, Topological properties of twisted cube, Inform. Sci. 113 (1999) 147–167.
[3] P. Cull, S.M. Larson, The Möbius cubes, IEEE Trans. Com-put. 44 (5) (1995) 647–659.
[4] K. Efe, A variation on the hypercube with lower diameter, IEEE Trans. Comput. 40 (11) (1991) 1312–1316.
[5] J. Fan, Hamilton-connectivity and cycle embedding of the Möbius cubes, Inform. Process. Lett. 82 (2002) 113–117. [6] J. Fan, X. Lin, X. Jia, Node-pancyclic and edge-pancyclic of
crossed cubes, Inform. Process. Lett. 93 (2005) 133–138. [7] J. Fan, N. Yu, X. Jia, X. Lin, Embedding of cycles in twisted
cubes with edge-pancyclic, Algorithmica, submitted for publica-tion.
[8] J. Fan, N. Yu, X. Jia, X. Lin, R.W.H. Lau, Edge-pancyclicity of twisted cubes, in: ISAAC’05, in: Lecture Notes in Computer Science, vol. 3827, Springer-Verlag, Berlin, 2005, pp. 1090– 1099.
[9] P.A.J. Hilbers, M.R.J. Koopman, J.L.A. van de Snepscheut, The twisted cube, in: PARLE’87, in: Lecture Notes in Computer Science, vol. 258/259, Springer-Verlag, Berlin, 1987, pp. 152– 159.
[10] M. Xu, J.M. Xu, Edge-pancyclicity of Möbius cubes, Inform. Process. Lett. 96 (2005) 136–140.
[11] X. Yang, D.J. Evans, G.M. Megson, The locally twisted cubes, Int. J. Comput. Math. 82 (2005) 401–413.
[12] X. Yang, G.M. Megson, D.J. Evans, Locally twisted cubes are 4-pancyclic, Appl. Math. Lett. 17 (2004) 919–925.
