DOI 10.1007/s11227-009-0317-2
Mutually independent Hamiltonian cycles in dual-cubes
Yuan-Kang Shih· Hui-Chun Chuang · Shin-Shin Kao· Jimmy J.M. Tan
Published online: 24 July 2009
© Springer Science+Business Media, LLC 2009
Abstract The hypercube family Qnis one of the most well-known interconnection
networks in parallel computers. With Qn, dual-cube networks, denoted by DCn, was
introduced and shown to be a (n+ 1)-regular, vertex symmetric graph with some fault-tolerant Hamiltonian properties. In addition, DCn’s are shown to be superior to
Qn’s in many aspects. In this article, we will prove that the n-dimensional dual-cube DCncontains n+1 mutually independent Hamiltonian cycles for n ≥ 2. More specif-ically, let vi∈ V (DCn)for 0≤ i ≤ |V (DCn)| − 1 and let v0, v1, . . . , v|V (DCn)|−1, v0
be a Hamiltonian cycle of DCn. We prove that DCncontains n+1 Hamiltonian cycles
of the formv0, v1k, . . . , v|V (DCk n)|−1, v0 for 0 ≤ k ≤ n, in which v k i = vk i whenever
k= k. The result is optimal since each vertex of DCnhas only n+ 1 neighbors.
Keywords Hypercube· Dual-cube · Hamiltonian cycle · Hamiltonian connected · Mutually independent
1 Introduction
An n-dimensional hypercube Qnis a graph with the vertex set{0, 1}nand there is an
edge between any two vertices that differ exactly in one bit position. The hypercube family is one of the most well-known and popular interconnection networks due to its excellent properties such as the recursive structure, symmetry, small diameter, low degree, easy routing, and so on; see [7,8,12,14,30].
Y.-K. Shih· J.J.M. Tan
Department of Computer Science, National Chiao Tung University, Hsinchu, 30010 Taiwan, China H.-C. Chuang· S.-S. Kao (
)Department of Applied Mathematics, Chung-Yuan Christian University, Chung-Li, 32023 Taiwan, China
The dual-cube family DCn, n≥ 1, was first introduced by Li, Peng, and Wu [17].
They make 2n+1copies of Q
nand divide them into two classes, Class 0 and Class 1.
Each class consists of 2n copies of Qn and each copy is called a cluster. By
prop-erly adding edges, they connect every pair of clusters from the opposite classes with an edge and prove that DCn is a (n+ 1)-regular, vertex symmetric graph that
con-tains some properties superior to hypercubes. Notice that the number of vertices of an n-dimensional dual-cube DCn is equal to the number of vertices of a (2n+
1)-dimensional hypercube Q2n+1. Each vertex in Q2n+1is adjacent to 2n+ 1 neighbors and the total number of edges of Q2n+1is (2n+ 1) × 22n. On the other hand, each vertex in DCn is adjacent to n+ 1 neighbors and the total number of edges of DCn
is (n+ 1) × 22n. Although any DCnhas much less edges than Q2n+1with the same
number of vertices, the diameter of DCn, 2n+ 2, is of the same order of the diameter
of Q2n+1, which is 2n+ 1. Other advanced subjects such as fault-tolerant cycle em-bedding and multiple disjoint paths construction in dual-cubes are also investigated [13,15–20].
The concept of mutually independent Hamiltonian cycles arises from the follow-ing application [22]. If k pieces of data must be sent from a message center u, and the data must be processed at each intermediate receiver (and the process takes time) before they are sent back to the center, then the existence of mutually independent cycles from u guarantees that there will be no waiting time for the parallel process-ing. Recently, many studies about mutually independent Hamiltonian cycles on hy-percubes and their variants are published [22–24]. In this article, we prove that the n-dimensional dual-cube DCncontains n+ 1 mutually independent Hamiltonian
cy-cles for n≥ 2. The result is optimal since DCnis a (n+ 1)-regular graph. The article is organized as follows. In Sect.2, we introduce the graph terminologies and nota-tions used in this paper, the precise definition of DCn and the new labeling of its
vertices. In Sect.3, we prove that DCn, n≥ 2, contains n + 1 mutually independent
Hamiltonian cycles.
2 Preliminaries
For the graph definitions and notations, we follow [3]. G= (V, E) is a graph if V is a finite set and E is a subset of{(u, v) | (u, v) is an unorder pair of V }. We say that V is the vertex set and E is the edge set of G. Two vertices u and v are adjacent if (u, v)∈ E. The total number of vertices of G is denoted by |V (G)|. For a vertex u of G, we denote the degree of u by deg(u)= |{v | (u, v) ∈ E}|. A graph G is k-regular if, for every vertex u∈ G, deg(u) = k.
A path is represented byv0, v1, v2, . . . , vk, where all vertices are distinct. We
also write the path v0, v1, v2, . . . , vk as v0, Q1, vi, vi+1, . . . , vj, Q2, vt, . . . , vk,
where Q1is the pathv0, v1, . . . , vi−1, vi and Q2is the pathvj, vj+1, . . . , vt−1, vt.
If a path P= v0, v1, v2, . . . , vk−1, vk, then P−1denotes the pathvk, vk−1, . . . , v2, v1, v0. A Hamiltonian path between u and v, where u and v are two distinct vertices of G, is a path joining u to v that visits every vertex of G exactly once. A graph G is
Fig. 1 The hypercubes Q1, Q2, and Q3
Hamiltonian connected if there exists a Hamiltonian path between any two different
vertices of G. Two paths P1= u0, u1, . . . , um and P2= v0, v1, . . . , vm from a to bare independent [22] if u0= v0= a, um= vm= b, and ui= vi for 1≤ i ≤ m − 1. Paths with the same number of vertices from a to b are mutually independent [22] if every two different paths are independent.
A graph G= (B ∪ W, E) is bipartite if V (G) is the union of two disjoint sets B and W and E⊆ {(u, v) | u ∈ B, v ∈ W}. It is easy to see that any bipartite graph with at least three vertices is not Hamiltonian connected. A bipartite graph G is
Hamil-tonian laceable if there exists a HamilHamil-tonian path between any two vertices from the
opposite partite sets.
A cycle is a path of at least three vertices such that the first vertex is the same as the last vertex. A Hamiltonian cycle of G is a cycle that visits every vertex of G exactly once. A Hamiltonian graph is a graph with a Hamiltonian cycle. The length of a cy-cle C is the number of edges/vertices in C. Two cycy-cles C1= u0, u1, . . . , uk, u0 and C2= v0, v1, . . . , vk, v0 beginning at s are independent if u0= v0= s and ui= vi for 1≤ i ≤ k [23]. Cycles beginning at s with equal length are mutually independent if every two different cycles are independent. Let G be a graph. We say that G
con-tains n mutually independent Hamiltonian cycles if there exist n Hamiltonian cycles
in G such that the n cycles begin at the same vertex s and are mutually independent. There are numerous studies in mutually independent Hamiltonian cycles. Readers can refer to [10,24,26,28].
An n-dimensional hypercube, denoted by Qn, is a graph with 2nvertices, 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. See Fig.1for an illustration. Sun et al. proved that the n-dimensional hypercube Qncontains n mutually independent Hamiltonian cycles for
n≥ 4 [28]. Other studies about hypercubes are in [4,5,9,12,21,25,29–31,33]. The dual-cube family DCn, n≥ 1, was first introduced by Li and Peng in
2000 [15]. Its nice structure has drawn the attention of many researchers [11,15–20]. A dual-cube DCn is obtained from a basic component Qn as follows. Make 2n+1
copies of Qnand divide them into two classes, Class 0 and Class 1. Each class
con-sists of 2ncopies of Qnand each copy is called a cluster. We shall label the 2n
clus-ters in each class by{0, 1}n, called the cluster id. Any vertex u∈ V (DCn)is given a
vertex id, which is a (2n+ 1)-bit binary string. Let u = u2nu2n−1· · · unun−1· · · u0.
If u2n= 0, then the next n bits u2n−1· · · unis called the cluster id and the last n bits un−1· · · u0is called the vertex id. If u2n= 1, then the next n bits u2n−1· · · unis called
Fig. 2 The graph DC2. Notice that label of each vertex u∈ V (DC2)consists of 5 bits. The first bit is the
class id. The two bits with the underline are called the cluster id, and the other two bits in Italic form are called the vertex id
the vertex id and the last n bits un−1· · · u0is called the cluster id. The following dia-gram gives an illustration.
u∈ Class 0: 0 Cluster id u2n−1u2n−2· · · un Vertex id in Qn un−1un−2· · · u0; u∈ Class 1: 1 Vertex id in Qn u2n−1u2n−2· · · un Cluster id un−1un−2· · · u0.
Given two vertices u= u2n· · · u0and v= v2n· · · v0, there is an edge between u and vin DCnif and only if the following conditions are satisfied:
• u and v differ in exactly one bit position i, where 0 ≤ i ≤ 2n; • if 0 ≤ i ≤ n − 1, then u2n= v2n= 0;
• if n ≤ i ≤ 2n − 1, then u2n= v2n= 1.
By the definition of DCn and the study of [20], we know that DCn is an (n+
1)-regular bipartite graph. Any vertex u in DCn is adjacent to n vertices in the same
cluster and to one vertex in some cluster of the other class. There is no edge between clusters of the same class. The edges within the same cluster are called regular-edges, and the edges connecting two clusters of distinct classes are called cross-edges. An illustration of DC2is given in Fig.2.
Fig. 3 Using the new labeling scheme which was proposed by Chen and Kao in [6] for V (DC2), we label
each vertex in DC2by (i, j, k), where i is the class id, j the cluster id, and k the vertex id
In 2008, Chen and Kao [6] proposed a more convenient new labeling for vertices of dual-cubes. Dual-cube DCn consists of two classes, Class 1 and Class 2. For i∈ {1, 2}, Class i has 2ncopies of Q
n, namely, Gi,1, . . . , Gi,2
n
, and each Gi,j is called a
cluster. For i∈ {1, 2}, let OGi= {Gi,j | 1 ≤ j ≤ 2nand j is odd} and EGi= {Gi,j |
1≤ j ≤ 2nand j is even}. Notice that each of OG1, OG2, EG1, and EG2consists of 2n−1 clusters. We shall label any vertex in Gi,j of DCn by (i, j, k), where k is the vertex id in Qn. Two vertices (i, j, k) and (i, j, k)are adjacent in DCn if and only
if one of the following conditions are satisfied:
(1) i= i, j= jand the vertices k and kare adjacent in Qn;
(2) |i − i| = 1, j = k, and k= j.
The edges satisfying (1) are regular-edges. The edges satisfying (2) are cross-edges, which connect different pairs of clusters belonging to the two classes. Vertices in a certain cluster use cross-edges to reach vertices in distinct clusters in the opposite class. Therefore, by regarding each cluster as a vertex, DCn becomes a complete
bipartite graph K2n,2n. Every cross-edge has the corresponding end vertices in the two
clusters of the opposite classes. For example, the cross-edge connecting the clusters G1,iand G2,j has end vertices (1, i, j )∈ G1,i and (2, j, i)∈ G2,j. Notice that DCn
is vertex symmetric. Figure3depicts DC2using the new labeling scheme mentioned above.
3 Mutually independent Hamiltonian cycles in dual-cubes
In this section, we use the new labeling scheme proposed in [6]. Throughout this section, we assume that vertex (i, j, k) is black (resp. white) in DCnif i+j +k is even
(resp. odd), without loss of generality. The following two results were established in [27] and [28].
Lemma 1 [27] The hypercube Qnis Hamiltonian laceable for any positive integer n.
Theorem 1 [28] The n-dimensional hypercube contains n− 1 mutually independent
Hamiltonian cycles for n∈ {1, 2, 3} and contains n mutually independent Hamil-tonian cycles for n≥ 4.
Assume that b= (i, j, k) is a black vertex, w = (3 − i, k, j) is a white vertex, and band w are connected by a cross-edge in DCn for i= 1, 2 and 1 ≤ j, k ≤ 2n. The
following results are true:
• If i = 1, j is odd and k is even, then b ∈ OG1
and w∈ EG2.
• If i = 1, j is even and k is odd, then b ∈ EG1and w∈ OG2.
• If i = 2, j is odd and k is odd, then b ∈ OG2and w∈ OG1.
• If i = 2, j is even and k is even, then b ∈ EG2and w∈ EG1. Therefore, we have the following property.
Property 1 In DCn, a black vertex in OG1, EG2, EG1,or OG2is adjacent to a white vertex in EG2, EG1, OG2,or OG1, respectively.
Theorem 2 The 2-dimensional dual-cube DC2 contains 3 mutually independent Hamiltonian cycles.
Proof Note that DC2is vertex symmetric. We assume that any Hamiltonian cycle be-gins at the vertex (1, 1, 1) without loss of generality. The three required cycles C1, C2, and C3beginning at (1, 1, 1) are constructed specifically. Please see AppendixA. Theorem 3 The 3-dimensional dual-cube DC3 contains 4 mutually independent Hamiltonian cycles.
Proof Since DC3is vertex symmetric, we assume that any Hamiltonian cycle begins at the vertex (1, 7, 1) without loss of generality. We construct the four required cycles C1, C2, C3, and C4beginning at (1, 7, 1). Please see AppendixB. Let ai’s and ai’s be clusters of DCn, where 1≤ i ≤ 2n+1. We say that two cluster
sequences S1= a1, a2, . . . , aj and S2= a1, a2, . . . , aj are independent if ai= ai for 1≤ i ≤ j. Cluster sequences with equal length are pairwise independent if any pair of the cluster sequences is independent. For any two positive integers r and d,
Lemma 2 For n≥ 2, there exist 2n−1− 1 pairwise independent cluster sequences of
the forma1, a2, . . . , a2n+1−1 in DCn−{G1,2
n−1
}, where a[i]4=0∈ OG1−{G1,2n−1}, a[i]4=1∈ EG2, a[i]4=2∈ EG1, and a[i]4=3∈ OG2.
Proof In DCn, there are 2n−1 clusters in each of EG1, EG2, OG1, and OG2. So OG1− {G1,2n−1} contains 2n−1− 1 clusters. We divide the cluster sequence a1, a2, . . . , a2n+1−1 into four subsequences. That is, S1= a1, a5, . . . , a2n+1−3,
S2= a2, a6, . . . , a2n+1−2, S3= a3, a7, . . . , a2n+1−1, and S4= a4, a8, . . . , a2n+1−4.
For 1≤ i ≤ 3, Si has 2n−1elements and there exist 2n−1choices for each element. Using the structure of a Latin square with 2n−1× 2n−1entries, we know that there exist 2n−1possible combinations of clusters in Si, denoted by ¯Sik for 1≤ k ≤ 2n−1,
such that{ ¯Sik}2kn−1=1 are pairwise independent cluster sequences.
Similarly, S4has 2n−1− 1 elements and we have 2n−1− 1 choices for each ele-ment. Hence, there exist 2n−1− 1 possible combinations of clusters in S4, denoted by ¯S4k for 1≤ k ≤ 2n−1− 1, such that { ¯S4k}2kn−1=1−1are pairwise independent cluster sequences.
Therefore, for n≥ 2, there exist (2n−1−1) pairwise independent cluster sequences
a1, a2, . . . , a2n+1−1 in DCn− {G1,2n−1}.
Lemma 3 Leta1, a2, . . . , a2n+1−1 be a cluster sequence of DCn−{G1,2n−1}, where
a[i]4=0∈ OG1− {G1,2n−1}, a[i]4=1∈ EG2, a[i]4=2∈ EG1, and a[i]4=3∈ OG2. Assume
that u is a white vertex in a1and v is a black vertex in a2n+1−1, then there is a
Hamil-tonian pathu = x1, H1, y1, x2, H2, y2, . . . , x2n+1−1, H2n+1−1, y2n+1−1= v between
uand v, where xi is a white vertex, yi is a black vertex,{xi, yi} ∈ V (ai), and Hi is a Hamiltonian path of ai joining xi to yifor every 1≤ i ≤ 2n+1− 1.
Proof By Property1, a black vertex yi in ai is adjacent to a white vertex xi+1in ai+1
by a cross-edge for 1≤ i ≤ 2n+1− 2. Notice that each cluster is a hypercube, u = x1 is a white vertex in a1and v= y2n+1−1is a black vertex in a2n+1−1. By Lemma 1, there is a Hamiltonian path Hi in cluster ai joining xi to yi for 1≤ i ≤ 2n+1− 1.
Thenu = x1, H1, y1, x2, H2, y2, . . . , x2n+1−1, H2n+1−1, y2n+1−1= v is the desired
Hamiltonian path.
Theorem 4 For n≥ 4, there are n + 1 mutually independent Hamiltonian cycles in
DCn.
Proof We want to construct n+ 1 mutually independent Hamiltonian cycles, denoted
by ¯Ci for 1≤ i ≤ n + 1, for DCn. Since DCn is vertex symmetric, without loss of
generality, we assume ¯Ci starts at (1, 2n− 1, 1) for 1 ≤ i ≤ n + 1. By Theorem1
and the fact that each cluster Gi,j in DC
n is Qn, there are n mutually independent
Hamiltonian cycles C1, C2, . . . , Cnbeginning at the white vertex (1, 2n− 1, 1) in the
cluster G1,2n−1. Without loss of generality, let Ci = v1∗, vi2, v3i, . . . , vi2n−1, vi2n, v1∗
for 1≤ i ≤ n, where v1∗= (1, 2n− 1, 1). Please see Fig.4for an illustration.
Notice that vmi is a white vertex if m is odd and vmi is a black vertex if m is even. Besides, 2n−1− 1 ≥ n + 1 when n ≥ 4. By Lemma2, we know that there exist
Fig. 4 An illustration for
Theorem4
2n−1− 1 pairwise independent cluster sequences for n ≥ 4. The 2n−1− 1 pairwise
independent cluster sequences are enough for us to construct the n+ 1 mutually independent Hamiltonian cycles below. There are three cases.
Case 1. To construct { ¯Ci : 1 ≤ i ≤ n − 1}. We consider the mutually independent Hamiltonian cycles C1, C2, . . . , Cn−1in G1,2
n−1
. Note that v2i is adjacent to a white vertex ¯vi2and v3i is adjacent to a black vertex ¯vi3by cross-edges for 1≤ i ≤ n − 1. By Lemma2, there exist n− 1 (≤ 2n−1− 1) pairwise independent cluster sequences
Ti= ai1, ai2, . . . , a2in+1−1 in DCn− {G1,2 n−1
} for 1 ≤ i ≤ n − 1 such that ¯vi
2∈ a
i
1 and¯vi3∈ a2in+1−1. By Lemma3, there is a Hamiltonian path Pi joining ¯v2i to¯vi3in Ti
for 1≤ i ≤ n − 1. Hence, ¯Ci= v1∗, v2i,¯vi2, Pi,¯vi3, vi3, . . . , vi2n, v1∗, 1 ≤ i ≤ n − 1, are
the desired cycles.
Case 2. To construct ¯Cn. In DCn, the black vertex vn2n is adjacent to the white
ver-tex ¯v2nnand the white vertex vn1is adjacent to the black ¯v1nby cross-edges. In DCn− {G1,2n−1}, by Lemma2, there is a cluster sequence T
n= a1n, a2n, . . . , a2nn+1−1, which
is pairwise independent with Ti for 1≤ i ≤ n−1 such that ¯v2nn∈ an1and¯v∗1∈ a2nn+1−1.
According to Lemma3, there is a Hamiltonian path Pnjoining¯vn2nto¯v1∗in Tn.
There-fore, ¯Cn= v1∗, vn2, . . . , v2nn,¯vn2n, Pn,¯v1∗, v1∗ is the desired cycle.
Case 3. To construct ¯Cn+1. The white vertex v∗1 is adjacent to the black vertex ¯v∗1
and the black vertex v2n is adjacent to the white vertex ¯v2n by cross-edges in DCn.
Because of Lemma2, there is a cluster sequence Tn+1= a1n+1, a
n+1 2 , . . . , a n+1 2n+1−1 in DCn− {G1,2 n−1
}, which is pairwise independent with Ti for 1≤ i ≤ n such that
¯v∗ 1∈ a
n+1
1 and ¯v2n∈ a
n+1
2n+1−1. By Lemma3, there is a Hamiltonian path Pn+1joining ¯v∗
1to ¯v2nin Tn+1. So ¯Cn+1= v∗1,¯v1∗, Pn+1,¯vn2, vn2, . . . , v2nn, v1∗ is the desired cycles.
Please see the Fig.5.
By Case 1, Case 2, and Case 3, the n+ 1 mutually independent Hamiltonian cy-cles ¯C1, ¯C2, . . . , ¯Cn+1in DCnfor n≥ 4 are constructed. This completes the proof.
Fig. 5 An illustration for Case 1, Case 2, and Case 3 in Theorem4. Notice that the gray areas are in
G1,2n−1
4 Conclusion
In [28], it was shown that the hypercube Qn, when n∈ {2, 3}, contains only n − 1
mutually independent Hamiltonian cycles. However, to our surprise, Theorem2and Theorem3show that DC2contains 3 mutually independent Hamiltonian cycles and
DC3contains 4 mutually independent Hamiltonian cycles. In addition, according to Theorem4, there are n+ 1 mutually independent Hamiltonian cycles in DCn for n≥ 4. Therefore, we have the following result.
Corollary 1 The n-dimensional dual-cube DCncontains n+1 mutually independent Hamiltonian cycles for n≥ 2.
Due to the fact that each vertex of DCnis connected to (n+ 1) vertices, there are
not any more Hamiltonian cycles emerging from the same vertex in addition to the n+ 1 mutually independent Hamiltonian cycles shown above. Therefore, our result is optimal. Another internet architecture that is more cost-effective (more scalable) than the traditional hypercubes and considered as a clever variation to the hypercube is the multi-ring [1,2,32]. Various studies such as the fault-tolerant cycle embedding and the existence of mutually independent Hamiltonian cycles in the multi-ring networks will be interesting topics to be explored.
Acknowledgements This research was partially supported by the National Science Council of the Re-public of China under contract NSC 96-2115-M-033-002-MY2.
Appendix A: The three required cycles of Theorem2
C1= (1, 1, 1), (1, 1, 2), (1, 1, 3), (1, 1, 4), (2, 4, 1), (2, 4, 2), (1, 2, 4), (1, 2, 1), (2, 1, 2), (2, 1, 3), (1, 3, 1), (1, 3, 2), (2, 2, 3), (2, 2, 4), (2, 2, 1), (2, 2, 2),
(1, 2, 2), (1, 2, 3), (2, 3, 2), (2, 3, 1), (2, 3, 4), (2, 3, 3), (1, 3, 3), (1, 3, 4), (2, 4, 3), (2, 4, 4), (1, 4, 4), (1, 4, 3), (1, 4, 2), (1, 4, 1), (2, 1, 4), (2, 1, 1), (1, 1, 1); C2= (1, 1, 1), (1, 1, 4), (2, 4, 1), (2, 4, 2), (2, 4, 3), (2, 4, 4), (1, 4, 4), (1, 4, 3), (1, 4, 2), (1, 4, 1), (2, 1, 4), (2, 1, 1), (2, 1, 2), (2, 1, 3), (1, 3, 1), (1, 3, 4), (1, 3, 3), (1, 3, 2), (2, 2, 3), (2, 2, 4), (2, 2, 1), (2, 2, 2), (1, 2, 2), (1, 2, 1), (1, 2, 4), (1, 2, 3), (2, 3, 2), (2, 3, 3), (2, 3, 4), (2, 3, 1), (1, 1, 3), (1, 1, 2), (1, 1, 1); C3= (1, 1, 1), (2, 1, 1), (2, 1, 4), (1, 4, 1), (1, 4, 2), (2, 2, 4), (2, 2, 3), (2, 2, 2), (2, 2, 1), (1, 1, 2), (1, 1, 3), (2, 3, 1), (2, 3, 2), (2, 3, 3), (2, 3, 4), (1, 4, 3), (1, 4, 4), (2, 4, 4), (2, 4, 3), (1, 3, 4), (1, 3, 3), (1, 3, 2), (1, 3, 1), (2, 1, 3), (2, 1, 2), (1, 2, 1), (1, 2, 2), (1, 2, 3), (1, 2, 4), (2, 4, 2), (2, 4, 1), (1, 1, 4), (1, 1, 1).
Appendix B: The four required cycles of Theorem3
C1= (1, 7, 1), (1, 7, 2), (1, 7, 3), (1, 7, 6), (1, 7, 7), (1, 7, 8), (1, 7, 5), (2, 5, 7), (2, 5, 2), (2, 5, 1), (2, 5, 8), (2, 5, 5), (2, 5, 4), (2, 5, 3), (2, 5, 6), (1, 6, 5), (1, 6, 4), (1, 6, 1), (1, 6, 8), (1, 6, 7), (1, 6, 2), (1, 6, 3), (1, 6, 6), (2, 6, 6), (2, 6, 3), (2, 6, 2), (2, 6, 7), (2, 6, 8), (2, 6, 1), (2, 6, 4), (2, 6, 5), (1, 5, 6), (1, 5, 3), (1, 5, 4), (1, 5, 5), (1, 5, 8), (1, 5, 1), (1, 5, 2), (1, 5, 7), (2, 7, 5), (2, 7, 4), (2, 7, 3), (2, 7, 6), (2, 7, 7), (2, 7, 2), (2, 7, 1), (2, 7, 8), (1, 8, 7), (1, 8, 2), (1, 8, 3), (1, 8, 6), (1, 8, 5), (1, 8, 4), (1, 8, 1), (1, 8, 8), (2, 8, 8), (2, 8, 1), (2, 8, 4), (2, 8, 5), (2, 8, 6), (2, 8, 7), (2, 8, 2), (2, 8, 3), (1, 3, 8), (1, 3, 5), (1, 3, 4), (1, 3, 3), (1, 3, 6), (1, 3, 7), (1, 3, 2), (1, 3, 1), (2, 1, 3), (2, 1, 6), (2, 1, 5), (2, 1, 4), (2, 1, 1), (2, 1, 8), (2, 1, 7), (2, 1, 2), (1, 2, 1), (1, 2, 8), (1, 2, 5), (1, 2, 4), (1, 2, 3), (1, 2, 6), (1, 2, 7), (1, 2, 2), (2, 2, 2), (2, 2, 7), (2, 2, 6), (2, 2, 3), (2, 2, 4), (2, 2, 5), (2, 2, 8), (2, 2, 1), (1, 1, 2), (1, 1, 1), (1, 1, 8), (1, 1, 7), (1, 1, 6), (1, 1, 5), (1, 1, 4), (1, 1, 3), (2, 3, 1), (2, 3, 2), (2, 3, 7), (2, 3, 8), (2, 3, 5), (2, 3, 6), (2, 3, 3), (2, 3, 4), (1, 4, 3), (1, 4, 6), (1, 4, 7), (1, 4, 2), (1, 4, 1), (1, 4, 8), (1, 4, 5), (1, 4, 4), (2, 4, 4),
(2, 4, 5), (2, 4, 8), (2, 4, 1), (2, 4, 2), (2, 4, 3), (2, 4, 6), (2, 4, 7), (1, 7, 4), (1, 7, 1); C2= (1, 7, 1), (1, 7, 4), (2, 4, 7), (2, 4, 2), (2, 4, 3), (2, 4, 6), (2, 4, 5), (2, 4, 8), (2, 4, 1), (2, 4, 4), (1, 4, 4), (1, 4, 1), (1, 4, 8), (1, 4, 7), (1, 4, 2), (1, 4, 3), (1, 4, 6), (1, 4, 5), (2, 5, 4), (2, 5, 5), (2, 5, 6), (2, 5, 3), (2, 5, 2), (2, 5, 1), (2, 5, 8), (2, 5, 7), (1, 7, 5), (1, 7, 8), (1, 7, 7), (1, 7, 2), (1, 7, 3), (1, 7, 6), (2, 6, 7), (2, 6, 6), (2, 6, 3), (2, 6, 4), (2, 6, 5), (2, 6, 8), (2, 6, 1), (2, 6, 2), (1, 2, 6), (1, 2, 5), (1, 2, 4), (1, 2, 3), (2, 3, 2), (2, 3, 7), (2, 3, 8), (2, 3, 1), (2, 3, 4), (2, 3, 5), (2, 3, 6), (2, 3, 3), (1, 3, 3), (1, 3, 6), (1, 3, 5), (1, 3, 4), (1, 3, 1), (1, 3, 8), (1, 3, 7), (1, 3, 2), (2, 2, 3), (2, 2, 6), (2, 2, 7), (2, 2, 2), (1, 2, 2), (1, 2, 1), (1, 2, 8), (1, 2, 7), (2, 7, 2), (2, 7, 3), (2, 7, 4), (2, 7, 1), (1, 1, 7), (1, 1, 8), (1, 1, 1), (1, 1, 4), (1, 1, 5), (1, 1, 6), (1, 1, 3), (1, 1, 2), (2, 2, 1), (2, 2, 4), (2, 2, 5), (2, 2, 8), (1, 8, 2), (1, 8, 3), (1, 8, 4), (1, 8, 1), (1, 8, 8), (1, 8, 5), (1, 8, 6), (1, 8, 7), (2, 7, 8), (2, 7, 7), (2, 7, 6), (2, 7, 5), (1, 5, 7), (1, 5, 2), (1, 5, 3), (1, 5, 6), (1, 5, 5), (1, 5, 4), (1, 5, 1), (1, 5, 8), (2, 8, 5), (2, 8, 8), (2, 8, 7), (2, 8, 2), (2, 8, 1), (2, 8, 4), (2, 8, 3), (2, 8, 6), (1, 6, 8), (1, 6, 5), (1, 6, 4), (1, 6, 3), (1, 6, 6), (1, 6, 7), (1, 6, 2), (1, 6, 1), (2, 1, 6), (2, 1, 3), (2, 1, 2), (2, 1, 1), (2, 1, 4), (2, 1, 5), (2, 1, 8), (2, 1, 7), (1, 7, 1); C3= (1, 7, 1), (1, 7, 8), (1, 7, 5), (1, 7, 4), (1, 7, 3), (1, 7, 6), (1, 7, 7), (2, 7, 7), (2, 7, 2), (2, 7, 3), (2, 7, 6), (2, 7, 5), (2, 7, 4), (2, 7, 1), (2, 7, 8), (1, 8, 7), (1, 8, 2), (1, 8, 3), (1, 8, 6), (1, 8, 5), (1, 8, 4), (1, 8, 1), (1, 8, 8), (2, 8, 8), (2, 8, 5), (2, 8, 4), (2, 8, 3), (2, 8, 6), (2, 8, 7), (2, 8, 2), (2, 8, 1), (1, 1, 8), (1, 1, 5), (1, 1, 4), (1, 1, 3), (1, 1, 6), (1, 1, 7), (1, 1, 2), (1, 1, 1), (2, 1, 1), (2, 1, 8), (2, 1, 6), (2, 1, 4), (2, 1, 3), (2, 1, 2), (2, 1, 7), (2, 1, 6), (1, 6, 1), (1, 6, 8), (1, 6, 7), (1, 6, 2), (1, 6, 3), (1, 6, 6), (1, 6, 5), (1, 6, 4), (2, 4, 6), (2, 4, 7), (2, 4, 2), (2, 4, 3), (2, 4, 4), (2, 4, 1), (2, 4, 8), (2, 4, 5), (1, 5, 4), (1, 5, 1), (1, 5, 8), (1, 5, 5), (1, 5, 6), (1, 5, 7), (1, 5, 2), (1, 5, 3), (2, 3, 5), (2, 3, 8), (2, 3, 1), (2, 3, 2), (2, 3, 7), (2, 3, 6), (2, 3, 3), (2, 3, 4), (1, 4, 3), (1, 4, 2), (1, 4, 7), (1, 4, 8), (1, 4, 1), (1, 4, 4), (1, 4, 5), (1, 4, 6), (2, 6, 4), (2, 6, 5), (2, 6, 8), (2, 6, 1), (2, 6, 2), (2, 6, 7), (2, 6, 6), (2, 6, 3), (1, 3, 6),
(1, 3, 7), (1, 3, 2), (1, 3, 3), (1, 3, 4), (1, 3, 1), (1, 3, 8), (1, 3, 5), (2, 5, 3), (2, 5, 4), (2, 5, 1), (2, 5, 8), (2, 5, 5), (2, 5, 6), (2, 5, 7), (2, 5, 2), (1, 2, 5), (1, 2, 8), (1, 2, 1), (1, 2, 4), (1, 2, 3), (1, 2, 6), (1, 2, 7), (1, 2, 2), (2, 2, 2), (2, 2, 3), (2, 2, 4), (2, 2, 1), (2, 2, 8), (2, 2, 5), (2, 2, 6), (2, 2, 7), (1, 7, 2), (1, 7, 1); C4= (1, 7, 1), (2, 1, 7), (2, 1, 6), (2, 1, 3), (2, 1, 4), (2, 1, 5), (2, 1, 8), (2, 1, 1), (2, 1, 2), (1, 2, 1), (1, 2, 8), (1, 2, 5), (1, 2, 4), (1, 2, 3), (1, 2, 6), (1, 2, 7), (1, 2, 2), (2, 2, 2), (2, 2, 7), (2, 2, 8), (2, 2, 1), (2, 2, 4), (2, 2, 5), (2, 2, 6), (2, 2, 3), (1, 3, 2), (1, 3, 7), (1, 3, 8), (1, 3, 1), (1, 3, 4), (1, 3, 5), (1, 3, 6), (1, 3, 3), (2, 3, 3), (2, 3, 6), (2, 3, 7), (2, 3, 2), (2, 3, 1), (2, 3, 8), (2, 3, 5), (2, 3, 4), (1, 4, 3), (1, 4, 2), (1, 4, 7), (1, 4, 8), (1, 4, 1), (1, 4, 4), (1, 4, 5), (1, 4, 6), (2, 6, 4), (2, 6, 1), (2, 6, 8), (2, 6, 7), (2, 6, 2), (2, 6, 3), (2, 6, 6), (2, 6, 5), (1, 5, 6), (1, 5, 3), (1, 5, 4), (1, 5, 1), (1, 5, 2), (1, 5, 7), (1, 5, 8), (1, 5, 5), (2, 5, 5), (2, 5, 8), (2, 5, 1), (2, 5, 4), (2, 5, 3), (2, 5, 2), (2, 5, 7), (2, 5, 6), (1, 6, 5), (1, 6, 6), (1, 6, 3), (1, 6, 4), (1, 6, 1), (1, 6, 2), (1, 6, 7), (1, 6, 8), (2, 8, 6), (2, 8, 7), (2, 8, 8), (2, 8, 5), (2, 8, 4), (2, 8, 3), (2, 8, 2), (2, 8, 1), (1, 1, 8), (1, 1, 1), (1, 1, 2), (1, 1, 3), (1, 1, 4), (1, 1, 5), (1, 1, 6), (1, 1, 7), (2, 7, 1), (2, 7, 4), (2, 7, 5), (2, 7, 6), (2, 7, 3), (2, 7, 2), (2, 7, 7), (2, 7, 8), (1, 8, 7), (1, 8, 6), (1, 8, 5), (1, 8, 8), (1, 8, 1), (1, 8, 2), (1, 8, 3), (1, 8, 4), (2, 4, 8), (2, 4, 1), (2, 4, 2), (2, 4, 3), (2, 4, 4), (2, 4, 5), (2, 4, 6), (2, 4, 7), (1, 7, 4), (1, 7, 5), (1, 7, 6), (1, 7, 3), (1, 7, 2), (1, 7, 7), (1, 7, 8), (1, 7, 1). References
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![Fig. 3 Using the new labeling scheme which was proposed by Chen and Kao in [ 6 ] for V (DC 2 ) , we label](https://thumb-ap.123doks.com/thumbv2/9libinfo/7685621.142655/5.659.79.584.71.442/fig-using-labeling-scheme-proposed-chen-kao-label.webp)
