2
Decomposition of Wheel Graphs into
Stars, Cycles and Paths: An Exposition
3
by
4
Norvie P. Agustin
1
5
6
7
8
9
An Undergraduate Seminar Paper presented and submitted to the
Department of Mathematics and Statistics
College of Science and Mathematics
Western Mindanao State University
Zamboanga City, Philippines 7000
10
In partial fulfillment of the requirements for the degree
11
Bachelor of Science in Mathematics
12
May 2023
13
Abstract
14
Decomposition of Wheel Graphs into Stars, Cycles and Paths: An
15
Exposition
16
by Norvie P. Agustin
17
This paper is an exposition of M. Subbulakshmi and I. Valliammal’s work on
18
“Decomposition of Wheel Graphs into Stars, Cycles and Paths”. We study in detailed the
19
decomposition of wheel graphs into claws, cycles and paths following some conditions for
20
such a decomposition to be exist.
21
theorem on how to decompose the wheel graph.
Also, we present some examples to illustrate the
Table of Contents
22
23
Title Page
i
24
Abstract
i
25
Table of Contents
ii
26
1 Introduction
1
27
2 Preliminaries
3
28
2.1
Basic Concepts in Number Theory . . . . . . . . . . . . . . . . . . . . . . . . .
3
29
2.2
Basic Concepts in Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . .
3
30
3 Main Results
10
31
4 Summary and Recommendation
34
32
4.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
33
4.2
Recommendation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
34
List of References
36
ii
35
Chapter 1
36
Introduction
37
“The origin of graph theory started with the problem of Konigsberg in the year 1735.
38
During recent decades, graph theory has developed into a major area of mathematics. It is
39
a delightful playground for the exploration of proof techniques in discrete mathematics and
40
its results have applications in many areas of computing, social and natural sciences.” [9]
41
“Graph theory is a significant component of Discrete Mathematics. Graph decomposition
42
is one of the broad area of research among the various elements of graph theory. A variety
43
of research articles and papers on graph decomposition have been published recently.” [9]
44
“Graph decomposition originated from combinatorial problems, the most of which
45
appeared in the 19th century. After the turn of the 19th century, the first publication
46
dealing directly with graph decomposition appeared.
47
decomposition has grown, with a significant increase after 1950. Graph decomposition is
48
now one of the most significant fields in graph theory and combinatorics. Many different
49
types of decomposition have been well researched in the literature. There are numerous
50
applications for graph decomposition, including group testing, DNA library screening,
51
scheduling issues, sharing schemes, and synchronous optical networks.” [9]
Since then, interest in graph
52
“Decomposition of graphs is one of the prominent areas of research in graph theory and
53
Combinatorial Design Theory. Various types of decomposition have been suggested by
54
different authors.” This paper focuses on decomposition of wheel graphs Wn based on work
55
of M. Subbulakshmi and I. Valliammal [10] entitled “Decomposition of Wheel Graphs into
1
Introduction
2
56
Stars, Cycles and Paths” and characterize the theorem of decomposition of wheel graph
57
into some other graphs such as claws, cycles and paths.
58
This paper organized as follows: In Chapter 2, we present some basic concepts needed
59
to comprehend the main topic of this study. In Chapter 3, we investigate and study in
60
detail their proofs, give illustrations about decomposition of wheel graphs into claws, cycles
61
and paths following some conditions for such a decomposition to be exist. And lastly, In
62
Chapter 4 will be the summary and recommendations.
63
Chapter 2
64
Preliminaries
65
This chapter presents some basic concepts needed in this study.
66
2.1
67
Definition 2.1. [3] “If n and d are integers and d ̸= 0, then n is divisible by d if and only
68
if n equal d times some integer k. The notation d | n is read as “d divides n”. Symbolically,
69
if n and d are integers and d ̸= 0: d | n ⇔ ∃ an integer k such that n = dk.”
70
Example 2.1. “Let n = 21 and d = 3. Then
Basic Concepts in Number Theory
71
3 | 21 since 21 = 3 · 7 where k = 7.”
72
Definition 2.2. [3] “Let m and n be integers and d be a positive integer. We say that m
73
is congruent to n modulo d and we write m ≡ n (mod d) if and only if d | (m − n).
74
Symbolically, m ≡ n (mod d) ⇔ d | (m − n).”
75
Example 2.2. “Let m = 12, n = 7 and d = 5. Since 12 − 7 = 5 is divisible by 5, that is
5 | (12 − 7)
76
77
and so,
12 ≡ 7 (mod 5).”
78
2.2
Basic Concepts in Graph Theory
79
Definition 2.3. [2] “A graph G consists of a finite non-empty set V (G) of elements called
80
vertices, and a finite set E(G) called edges. We call V (G) the vertex set and E(G) the
3
Preliminaries
4
81
edge set of G. If e is an edge and u and v are vertices such that uv ∈ E(G), then e is said
82
to join u and v; the vertices u and v are called the endpoints of e.”
83
Example 2.3. “Consider a graph below. Graph G has a vertex set {u, v, w, x, y} and edge
set {uv, uw, ux, vx, vw, wx, xy}.”
v
w
y
u
x
Figure 2.1: Graph G
84
85
Definition 2.4. [2] “A graph H is a subgraph of G (written H ⊆ G) if V (H) ⊆ V (G) and
86
E(H) ⊆ E(G) and the assignment of endpoints to edges in H are the same as in G.”
87
Example 2.4. “Consider the two graphs G and H with V (G) = {u, v, w, x, y}, V (H) =
88
{v, w, x} and E(G) = {uv, vw, vx, wx, xy}, E(H) = {vw, vx, wx}. Since V (H) ⊆ V (G)
and E(H) ⊆ E(G), Hence, H is a subgraph of G.”
v
w
v
G:
w
H:
u
x
y
x
Figure 2.2: Graph G and Subgraph H
89
90
Definition 2.5. [5] “Let S ⊆ V (G), S ̸= ∅. The induced subgraph, denoted by ⟨S⟩, is
91
the maximal subgraph of G with vertex set S. Thus, two vertices of S are adjacent in ⟨S⟩
92
if and only if they are adjacent to G. A subgraph H is called a vertex-induced subgraph
93
(or simply, induced subgraph) of G if H = ⟨S⟩ for some S ⊆ V (G).”
Preliminaries
5
94
Definition 2.6. [5] “Let E ⊆ E(G), E ̸= ∅. The subgraph induced by E, denoted by ⟨E⟩,
95
is the minimal subgraph of G with edge set E. A subgraph H is called a edge-induced
96
subgraph of G if H = ⟨E⟩ for some E ⊆ E(G).”
97
Example 2.5. “Consider a graph G in Figure 2.2. H1 is the induced subgraph of G with
S = {u, w, x, y} and H2 is the edge-induced subgraph of G with E = {uv, vw}.”
w
H1 :
v
H2 :
u
w
u
y
x
Figure 2.3: Induced Subgraph and Edge-induced Subgraph of G
98
99
Definition 2.7. [5] “A path denoted by Pn , n ≥ 2 is a graph with vertex set {v1 , v2 , . . . , vn }
100
and edge set {v1 v2 , v2 v3 , . . . , vn−1 vn } where vn are distinct. The length of Pn is said to be
101
n − 1.”
102
Example 2.6. “Consider a path P6 below.
A path P6 of length 5 has vertex set
{v1 , v2 , v3 , v4 , v5 , v6 } and edge set {v1 v2 , v2 v3 , v3 v4 , v4 v5 , v5 v6 }.”
v1
v2
v3
v4
v5
v6
Figure 2.4: Path Graph P6
103
104
Definition 2.8. [5] “A cycle of length n denoted by Cn , n ≥ 3 is a graph with vertex set
105
{v1 , v2 , . . . , vn , v1 } and edge set {v1 v2 , v2 v3 , . . . , vn−1 vn , vn v1 } where vn are distinct.”
106
Example 2.7. “Consider a cycle C8 below. A cycle C8 of length 8 has a vertex set
107
{v1 , v2 , v3 , v4 , v5 , v6 , v7 , v8 } and edge set {v1 v2 , v2 v3 , v3 v4 , v4 v5 , v5 v6 , v6 v7 , v7 v8 , v8 v1 }.”
Preliminaries
6
v1
v2
v8
v3
v7
v4
v6
v5
Figure 2.5: Cycle Graph C8
108
Definition 2.9. [2] “A graph in which pair of distinct vertices is joined by an edge is called
109
a complete graph. A complete graph with n vertices is denoted by Kn .”
110
Example 2.8. “Figure 2.6 shows a complete graph K5 with vertex set {v1 , v2 , v3 , v4 , v5 }
where each vertex is adjacent to each other vertex.”
v1
v5
v2
v4
v3
Figure 2.6: Complete Graph K5
111
112
Definition 2.10. [2] “A bipartite graph is a graph whose vertex set can be partitioned
113
into two subsets X and Y so that each edge has one end in X and one end in Y ; such a
114
partition (X, Y ) is called a bipartition of the graph.”
Example 2.9. “Figure 2.7 shows the example of bipartite graph G.”
Figure 2.7: Bipartite Graph G
115
Preliminaries
7
116
Definition 2.11. [2] “A complete bipartite graph is a bipartite graph with bipartition
117
(X, Y ) in which each vertex of X is joined to each vertex of Y ; if X and Y have m and n
118
vertices, we write Km,n .”
119
Example 2.10. “Figure 2.8 shows a complete bipartite K4,3 where set X consist of vertex
120
set {a, b, c, d} and set Y consist of vertex set {x, y, z}. That is, every vertex of set X is
adjacent to every vertex of set Y.”
a
c
b
d
y
x
z
Figure 2.8: Complete Bipartite K4,3
121
122
Definition 2.12. [10] “A star , denoted by Sn , is a complete bipartite K1,n with one
123
internal vertex and n edges. A star with 3 edges, denoted by S3 , is called a claw .”
124
Example 2.11. “Figure 2.9 shows a star S3 where internal vertex a is adjacent to 3 vertices
x, y and z.”
x
a
z
y
Figure 2.9: Star Graph S3
125
126
Definition 2.13. [10] “A wheel graph of length 2n, denoted by Wn is a graph with vertex
127
set {v0 , v1 , v2 , . . . , vn } and edge set containing of all edges of the form vi vi+1 and v0 v1 where
128
1 ≤ i ≤ n. Wn has every vertex of degree 3 except the internal vertex which has degree n.”
Preliminaries
8
129
Example
2.12. “Figure
2.10
shows
a
wheel
graph
W6
with
vertex
set
130
{v0 , v1 , v2 , v3 , v4 , v5 , v6 } where each vertex has degree 3 except for internal vertex which has
degree 6.”
v1
v6
v2
v3
v0
v5
v4
Figure 2.10: Wheel Graph W6
131
132
Definition 2.14. [2] “Two graphs G and H are isomorphic (written G ∼
= H or sometimes
133
G = H) if there is bijection f : V (G) → V (H) such that uv ∈ E(G) if and only if
134
f (u)f (v) ∈ E(H). If G and H are isomorphic, we say “G is isomorphic to H”.”
135
Example 2.13. “Consider the graph G with V (G) = {a, b, c, d, e} and graph H with
136
V (H) = {1, 2, 3, 4, 5}. Since a = f (1), b = f (2), c = f (3), d = f (4), and e = f (5), hence
G∼
= H.”
e
b
3
c
G:
a
d
2
H:
4
1
5
Figure 2.11: Isomorphic Graphs G and H
137
138
Definition 2.15. [2] “The union of graph G1 , . . . , Gk is the subgraph with the vertex set
139
Sk
i=1 V
(Gi ) and edge set
Sk
i=1 E(Gi ),
written G1 ∪ . . . ∪ Gk .”
Preliminaries
9
140
Definition 2.16. [1] “A decomposition of a graph G denoted by D(G), is a collection of
141
edge-disjoint subgraphs H1 , H2 , . . . , Hr of G such that every edge of G belongs to exactly
142
one Hi .”
143
Definition 2.17. [10] “Let L = {H1 , H2 , . . . , Hr } be a family of subgraphs of G. An
144
L − decomposition of G is an edge-disjoint decomposition of G into positive integer αi
145
copies of Hi , where i ∈ {1, 2, . . . , r}. Furthermore, if Hi (i ∈ {1, 2, . . . , r}) is isomorphic to
146
a graph H, then we say that G has an H − decomposition.”
147
Example 2.14. “Consider the graph G below. Then G can be decomposed into 3 copies of
path P3 .
x
y
v
z
w
Figure 2.12: Graph G decomposed into three copies of P3
148
Thus, we can take L = {H1 , H2 , H3 } as L − decomposition of G as shown below.
y
H1
x
x
z
z
v
w
H2
z
w
H3
Figure 2.13: L = {H1 , H2 , H3 }, where E(H1 ) ∪ E(H2 ) ∪ E(H3 ) = E(G)
149
150
Moreover, {H1 , H2 , H3 } is also an H − decomposition of G since H1 ∼
= H2 ∼
= H3 .”
151
Chapter 3
152
Main Results
153
In this section, we discussed in detail the paper of M. Subbulakshmi and I. Valliammal
154
on “Decomposition of Wheel Graphs into Stars, Cycles and Paths” [10].
155
“Decomposition of wheel graphs into claws, cycles and paths”
156
Theorem 3.1. “Any wheel graph Wn , n ≥ 3 can be decomposed into Cn and Sn .”
157
Proof. For n ≥ 3, let Wn be a wheel graph. To form a Wn , we start from cycle with n
158
vertices and label each vertex from v1 , v2 , . . . , vi for i = 1, 2, . . . , n. Note that each vertex
159
of a cycle Cn has exactly two edges incident with it, represent the outer vertices of the Wn .
160
Next, we create a central vertex and join it to each vertex of the cycle Cn . Note that the
161
central vertex labeled as v0 . Hence, we have the resulting graph of Wn .
162
Now, observe that v0 and vi are inner and outer vertex of Wn , respectively. Notice that
163
v0 is adjacent to each vertex vi for i = 1, 2, . . . , n, that is v0 v1 , v0 v2 , . . . , v0 vn . Also, vi is
164
adjacent to vi+1 for i = 1, 2, . . . , n and vn is adjacent to v1 , that is v1 v2 , v2 v3 , . . . , vn v1 . This
165
means that the subgraph induced by edge set {v0 v1 , v0 v2 , . . . , v0 vn } forms a star Sn and
166
subgraph induced by edge set {v1 v2 , v2 v3 , . . . , vn v1 } forms a cycle Cn . Thus, Wn can be
167
decomposed into Cn and Sn .
168
Example 3.1. Consider a wheel graph W3 with vertex set {v0 , v1 , v2 , v3 }. W3 is decomposed
169
by taking the subgraph induced by edge set {v1 v2 , v2 v3 , v3 v1 } forms a cycle C3 and subgraph
10
Main Results
170
11
induced by edge set {v0 v1 , v0 v2 , v0 v3 } forms a claw S3 . Hence, W3 decomposed into C3 and
S3 .
v1
v1
v0
v3
v1
v0
=⇒
v2
v3
v2
v3
C3
W3
v2
S3
Figure 3.1: Decomposition of W3 into C3 and S3
171
Theorem 3.2. “Any wheel graph Wn , n ≥ 3 can be decomposed into following ways.” “
D(Wn ) =




(n − 2d)S3 , d = 1, 2, 3, . . .
if n ≡ 0 (mod 6)









[(n − 2d) − 1] S3 and P3 , d = 1, 2, 3, . . . if n ≡ 1 (mod 6)







 [(n − 2d) − 1] S3 and P2 , d = 1, 2, 3, . . . if n ≡ 2 (mod 6)



(n − 2d)S3 and C3 , d = 1, 2, 3, . . .









(n − 2d)S3 and P3 , d = 1, 2, 3, . . .







 (n − 2d)S3 and P2 , d = 1, 2, 3, . . .
if n ≡ 3
(mod 6)
if n ≡ 4
(mod 6)
if n ≡ 5
(mod 6)
.
172
”
173
Proof. Let Wn be any wheel graph of order n ≥ 3 with vertex set {v0 , v1 , v2 , . . . , vn } and
174
edge set {e1 , e2 , . . . , en , e′1 , e′2 , . . . , e′n } where ei and e′i , for 1 ≤ i ≤ n are the edges in outer
175
and inner cycles, respectively. Notice that vi and vi+1 are the endpoints of ei and also v0
176
and vi are the endpoints of e′i . Then consider the six cases below:
177
Case 1. n ≡ 0 (mod 6).
Main Results
12
178
By Definition 2.2, observe that n ≡ 0 (mod 6) if and only if 6 | n − 0, then n = 6d for
179
d = 1, 2, 3, . . .. We show that Wn can be decomposed into (n − 2d)S3 that is, Wn can be
180
decomposed into (n − 2d)S3 = (6d − 2d)S3 = 4dS3 .
181
Consider the edge sets
182
Ei = {ei , e′i+1 , ei+1 }, where i = 1, 3, 5, . . . , n − 1 and
183
Ej = {e′i , e′i+2 , e′i+4 }, where i = 1, 7, 13, . . . , n − 5.
Observe that the set of values of i for Ei is arithmetic sequence whereas every term is
obtained by adding constant number 2 to its previous term. Since the sequence is ranging
from 1 to n − 1, that is i can be express as 2k − 1 for k = 1, 2, 3, . . .. Simplifying this
expression, we get
i = n − 1 =⇒ 2k − 1 = 6d − 1
2k = 6d
k = 3d
where k is no. of ⟨Ei ⟩.
Similarly, the set of values of i for Ej is arithmetic sequence of constant number 6. Since
the sequence is ranging from 1 to n−5, that is i can be express as 6k−5 where k = 1, 2, 3, . . ..
Simplifying this expression, we get
i = n − 5 =⇒ 6k − 5 = 6d − 5
6k = 6d
k= d
where k is no. of ⟨Ej ⟩.
Main Results
13
184
Thus, the edge-induced subgraph ⟨Ei ⟩ forms 3d claws that consists of two outer edges
185
and one inner edge incident to a common vertex vi+1 for i = 1, 3, 5, . . . , n − 1 and the
186
edge-induced subgraph ⟨Ej ⟩ forms d claws that consists of three inner edges incident to a
187
common vertex v0 . Therefore, Wn decomposed into 3d + d = 4d claws S3 .
188
Case 2. n ≡ 1 (mod 6).
189
By Definition 2.2, note that n ≡ 1 (mod 6) if and only if 6 | n − 1, then n = 6d + 1 for
190
d = 1, 2, 3, . . .. To show that Wn can be decomposed into [(n − 2d) − 1]S3 and P3 that is,
191
it is enough to show that Wn can decomposed into
[(n − 2d) − 1]S3 = [(6d + 1 − 2d) − 1]S3 = 4dS3 and P3 .
192
193
Consider the following edge sets
194
Ei = {ei , e′i+1 , ei+1 } where i = 1, 3, 5, . . . , n − 2
195
Ej = {e′i , e′i+2 , e′i+4 } where i = 3, 9, 15, . . . , n − 4 and
196
Ek = {e′1 , en }.
197
Observe that the set of values of i for Ei is arithmetic sequence whereas every term is
198
obtained by adding constant number 2 to its previous term. Since the sequence is ranging
199
from 1 to n − 2, that is i can be express as 2k − 1 for k = 1, 2, 3, . . .. Simplifying this
200
expression, we get
Main Results
14
i = n − 2 =⇒ 2k − 1 = (6d + 1) − 2
2k − 1 = 6d − 1
2k = 6d
k = 3d
where k is no. of ⟨Ei ⟩.
Also, the set of values of i for Ej is arithmetic sequence of constant number 6. Since the
sequence is ranging from 3 to n − 5, that is i can be express as 6k − 3 where k = 1, 2, 3, . . ..
Simplifying this expression, we get
i = n − 4 =⇒ 6k − 3 = (6d + 1) − 4
6k − 3 = 6d − 3
6k = 6d
k = d
where k is no. of ⟨Ej ⟩.
201
Which means that the edge-induced subgraph ⟨Ei ⟩ forms 3d claws that consists of two
202
outer edges and one inner edge incident to a common vertex vi+1 for i = 1, 3, 5, . . . , n−2 and
203
the edge-induced subgraph ⟨Ej ⟩ forms d claws that consists of three inner edges incident
204
to a common vertex v0 . Moreover, the edge induced subgraph ⟨Ek ⟩ forms a path P3 that
205
consist of two edges that are incident to vertex v1 . Hence, Wn can be decomposed into
206
3d + d = 4d claws S3 and path P3 .
207
Case 3. n ≡ 2 (mod 6).
Main Results
15
208
By Definition 2.2, note that n ≡ 2 (mod 6) if and only if 6 | n − 2, then n = 6d + 2 for
209
d = 1, 2, 3, . . .. We show that Wn can be decomposed into [(n − 2d) − 1]S3 and P2 that is,
210
it is enough to show that Wn can decomposed into
[(n − 2d) − 1]S3 = [(6d + 2 − 2d) − 1]S3 = (4d + 1)S3 and P2 .
211
212
Consider the following edge sets
213
Ei = {ei , e′i+1 , ei+1 }, where i = 1, 3, 5, . . . , n − 1
214
Ej = {e′i , e′i+2 , e′i+4 }, where i = 1, 7, 13, . . . , n − 7 and
215
Ek = {e′n−1 }.
Observe that the set of values of i for Ei is arithmetic sequence whereas every term is
obtained by adding constant number 2 to its previous term. Since the sequence is ranging
from 1 to n − 1, that is i can be express as 2k − 1 for k = 1, 2, 3, . . .. Simplifying this
expression, we get
i = n − 1 =⇒ 2k − 1 = (6d + 2) − 1
2k − 1 = 6d + 1
2k = 6d + 2
k = 3d + 1
where k is no. of ⟨Ei ⟩.
Similarly, the set of values of i for Ej is arithmetic sequence of constant number 6. Since
the sequence is ranging from 1 to n−7, that is i can be express as 6k−5 where k = 1, 2, 3, . . ..
Main Results
16
Simplifying this expression, we get
i = n − 7 =⇒ 6k − 5 = (6d + 2) − 7
6k − 5 = 6d − 5
6k = 6d
k = d
where k is no. of ⟨Ej ⟩.
216
This means that the edge-induced subgraph ⟨Ei ⟩ forms 3d + 1 claws that consists of two
217
outer edges and one inner edge incident to a common vertex vi+1 for i = 1, 3, 5, . . . , n − 1,
218
and the edge-induced subgraph ⟨Ej ⟩ forms d claws that consists of three inner edges incident
219
to a common vertex v0 . Also, the edge induced subgraph ⟨Ek ⟩ forms a path P2 with only
220
one inner edge. Therefore, Wn can be decomposed into 3d + 1 + d = 4d + 1 claws S3 and
221
path P2 .
222
Case 4. n ≡ 3 (mod 6).
223
By Definition 2.2, we know that n ≡ 3 (mod 6) if and only if 6 | n − 3, then n = 6k + 3 for
224
some positive integer k. We want to express n as 6d − 3 for some positive integer d. That is,
225
6d − 3 = 6k + 3 =⇒ 6d = 6k + 6 =⇒ d = k + 1 where k + 1 is an integer for some positive
226
integer k. Note that 6d − 3 can be written as 6k + 3 that is, 6d − 3 = 6(k + 1) − 3 = 6k + 3.
227
Thus, we have shown that if n ≡ 3 (mod 6), then n = 6d − 3 for d = 1, 2, 3, . . ..
228
229
230
Now we show that Wn can be decomposed into (n − 2d)S3 and C3 that is, Wn can
decomposed into (n − 2d)S3 = (6d − 3 − 2d)S3 = (4d − 3)S3 and C3 .
Consider the following edge sets
Main Results
17
231
Ei = {ei , e′i+1 , ei+1 }, where i = 1, 3, 5, . . . , n − 2
232
Ej = {e′i , e′i+2 , e′i+4 }, where i = 3, 9, 15, . . . , n − 6 and
233
Ek = {e′1 , e′n , en }.
Observe that the set of values of i for Ei is arithmetic sequence whereas every term is
obtained by adding constant number 2 to its previous term. Since the sequence is ranging
from 1 to n − 2, that is i can be express as 2k − 1 for k = 1, 2, 3, . . .. Simplifying this
expression, we get
i = n − 2 =⇒ 2k − 1 = (6d − 3) − 2
2k − 1 = 6d − 5
2k = 6d − 4
k = 3d − 2
where k is no. of ⟨Ei ⟩.
Moreover, the set of values of i for Ej is arithmetic sequence of constant number 6. Since
the sequence is ranging from 3 to n−6, that is i can be express as 6k−3 where k = 1, 2, 3, . . ..
Simplifying this expression, we get
i = n − 6 =⇒ 6k − 3 = (6d − 3) − 6
6k − 3 = 6d − 9
6k = 6d − 6
k = d−1
where k is no. of ⟨Ej ⟩.
Main Results
18
234
Therefore, the edge-induced subgraph ⟨Ei ⟩ forms 3d − 2 claws that consists of two outer
235
edges and one inner edge incident to a common vertex vi+1 for i = 1, 3, 5, . . . , n − 2, and the
236
edge-induced subgraph ⟨Ej ⟩ forms d − 1 claws that consists of three inner edges incident to
237
a common vertex v0 . Also, the edge induced subgraph ⟨Ek ⟩ forms a cycle C3 having two
238
internal edges and outer edge that every vertex v0 , v1 and vn has exactly two edges incident
239
with it. Hence, Wn can be decomposed into 3d − 2 + d − 1 = 4d − 3 claws S3 and cycle C3 .
240
Case 5. n ≡ 4 (mod 6).
241
By Definition 2.2, we note that n ≡ 4 (mod 6) if and only if 6 | n − 4, then n = 6k + 4
242
for some positive integer k. We want to express n as 6d − 2 for some integer d. That is,
243
6d − 2 = 6k + 4 =⇒ 6d = 6k + 6 =⇒ d = k + 1 where k + 1 is an integer for some positive
244
integer k. Note that 6d − 2 can be written as 6k + 4 that is, 6d − 2 = 6(k + 1) − 2 = 6k + 4.
245
Thus, we have shown that if n ≡ 4 (mod 6), then n = 6d − 2 for d = 1, 2, 3, . . ..
246
247
248
Next, we show that Wn can be decomposed into (n − 2d)S3 and P3 . That is, Wn can
decomposed into (n − 2d) = (6d − 2 − 2d) = (4d − 2)S3 and P3 .
Consider the following edge sets
249
Ei = {ei , e′i+1 , ei+1 }, where i = 1, 3, 5, . . . , n − 1
250
Ej = {e′i , e′i+2 , e′i+4 }, where i = 3, 9, 15, . . . , n − 7 and
251
Ek = {e′1 , e′n−1 }.
Observe that the set of values of i for Ei is arithmetic sequence whereas every term is
obtained by adding constant number 2 to its previous term. Since the sequence is ranging
from 1 to n − 1, that is i can be express as 2k − 1 for k = 1, 2, 3, . . .. Simplifying this
Main Results
19
expression, we get
i = n − 1 =⇒ 2k − 1 = (6d − 2) − 1
2k − 1 = 6d − 3
2k = 6d − 2
k = 3d − 1
where k is no. of ⟨Ei ⟩.
Similarly, the set of values of i for Ej is arithmetic sequence of constant number 6. Since
the sequence is ranging from 3 to n−7, that is i can be express as 6k−3 where k = 1, 2, 3, . . ..
Simplifying this expression, we get
i = n − 7 =⇒ 6k − 3 = (6d − 2) − 7
6k − 3 = 6d − 9
6k = 6d − 6
k = d−1
where k is no. of ⟨Ej ⟩.
252
Thus, the edge-induced subgraph ⟨Ei ⟩ forms 3d − 1 claws that consists of two outer edges
253
and one inner edge incident to a common vertex vi+1 for i = 1, 3, 5, ..., n − 1, and the edge-
254
induced subgraph ⟨Ej ⟩ forms d − 1 claws that consists of three inner edges incident to a
255
common vertex v0 . Moreover, the edge induced subgraph ⟨Ek ⟩ forms a path P3 consists
256
of inner and outer edge that are incident with v0 . Therefore, Wn can be decomposed into
257
3d − 1 + d − 1 = 4d − 2 claws S3 and path P3 .
258
Case 6. n ≡ 5 (mod 6).
Main Results
20
259
By Definition 2.2, we know that n ≡ 5 (mod 6) if and only if 6 | n − 5, then n = 6k + 5 for
260
some positive integer k. We want to express n as 6d − 1 for some positive integer d. That is,
261
6d − 1 = 6k + 5 =⇒ 6d = 6k + 6 =⇒ d = k + 1 where k + 1 is an integer for some positive
262
integer k. We show that 6d−1 can be written as 6k +5 that is, 6d−1 = 6(k +1)−1 = 6k +5.
263
Thus, we have shown that if n ≡ 5 (mod 6), then n = 6d − 1 for d = 1, 2, 3, . . ..
264
265
266
Now we show that Wn can be decomposed into (n − 2d)S3 and P2 . That is, Wn can
decomposed into (n − 2d)S3 = (6d − 1 − 2d)S3 = (4d − 1)S3 and P2 .
Consider the following edge sets
267
Ei = {ei , e′i+1 , ei+1 }, where i = 1, 3, 5, . . . , n − 2
268
Ej = {e′i , e′i+2 , e′i+4 }, where i = 1, 7, 13, . . . , n − 4 and
269
Ek = {en }.
Observe that the set of values of i for Ei is arithmetic sequence whereas every term is
obtained by adding constant number 2 to its previous term. Since the sequence is ranging
from 1 to n − 2, that is i can be express as 2k − 1 for k = 1, 2, 3, . . .. Simplifying this
expression, we ge
i = n − 2 =⇒ 2k − 1 = (6d − 1) − 2
2k − 1 = 6d − 3
2k = 6d − 2
k = 3d − 1
where k is no. of ⟨Ei ⟩.
Main Results
21
Also, the set of values of i for Ej is arithmetic sequence of constant number 6. Since the
sequence is ranging from 1 to n − 4, that is i can be express as 6k − 5 where k = 1, 2, 3, . . ..
Simplifying this expression, we get
i = n − 4 =⇒ 6k − 5 = (6d − 1) − 4
6k − 5 = 6d − 5
6k = 6d
where k is no. of ⟨Ej ⟩.
k = d
270
Hence, the edge-induced subgraph ⟨Ei ⟩ forms 3d − 1 claws that consists of two outer
271
edges and one inner edge incident to a common vertex vi+1 for i = 1, 3, 5, . . . , n − 2 and the
272
edge-induced subgraph ⟨Ej ⟩ forms d claws that consists of three inner edges incident to a
273
common vertex v0 . Moreover, the edge induced subgraph ⟨Ek ⟩ forms a path P2 of length 1.
274
Therefore, Wn can be decomposed into 3d − 1 + d = 4d − 1 claws S3 and path P2 .
275
276
Example 3.2. Consider the following illustrations for each cases below:
277
Case 1. If n ≡ 0 (mod 6), then n = 6d for d = 1, 2, 3, . . ..
278
For d = 1, then n = 6 since 6 ≡ 0 (mod 6) such that 6 | 6 − 0.
the
wheel
graph
W6
with
vertex
set
{v0 , v1 , . . . , v6 }
279
Consider
and
edge
set
280
{e1 , e2 , . . . , e6 , e′1 , e′2 , . . . , e′6 }. We show that W6 can be decomposed into (n − 2d)S3 , that is
281
W6 can be decomposed into n − 2d = 6d − 2d = 6(1) − 2(1) = 4 S3 .
Main Results
22
v2
e1
e2
v1
e6
v0
e′6
e′5
v6
v3
e′2
e′1
e5
e′3
e3
e′4
v4
e4
v5
Figure 3.2: Wheel Graph W6
282
Consider the edge sets Ei = {ei , e′i+1 , ei+1 }, i = 2k − 1, . . . , 5 for k = 1, 2, 3, . . . that is,
283
i = 2(1) − 1 = 1 =⇒ E1 = {e1 , e′2 , e2 }
284
i = 2(2) − 1 = 3 =⇒ E2 = {e3 , e′4 , e4 }
285
i = 2(3) − 1 = 5 =⇒ E3 = {e5 , e′6 , e6 }.
286
Similarly, Ej = {e′i , e′i+2 , e′i+4 }, i = 6k − 5, . . . , 1 for k = 1, 2, 3, . . . that is,
i = 6(1) − 5 = 1 =⇒ E1 = {e′1 , e′3 , e′5 }.
287
v2
e1
e2
v1
v3
e′2
v1
v1
v3
v0
v3
e′1
v0
v0
e6
e′3
e3
e′6
v0
e′5
e′4
v6
v4
e5
e4
v5
v5
v5
Figure 3.3: Decomposition of W6 into 4 copies of claw S3
288
289
That is, edge-induced subgraph ⟨Ei ⟩ forms 3 copies of claw S3 and ⟨Ej ⟩ forms a claw S3 .
Hence, W6 decomposed into four copies of claw S3 .
Main Results
290
291
23
Case 2. If n ≡ 1 (mod 6), then 6d + 1 for d = 1, 2, 3, . . ..
For d = 1, then n = 7 since 7 ≡ 1 (mod 6) such that 6 | 7 − 1.
the
wheel
graph
W7
with
vertex
{v0 , v1 , . . . , v7 }
292
Consider
set
and
edge
set
293
{e1 , e2 , . . . , e7 , e′1 , e′2 , . . . , e′7 }. We show that W7 can be decomposed into [(n − 2d) − 1]S3
294
and P3 , that is W7 can be decomposed into n − 2d = (6d + 1 − 2d) − 1 =
295
[6(1) + 1 − 2(1)] − 1 = 4 S3 and P3 .
v2
e1
v1
e2
v3
e′1
e′2
e7
e′3
e′7
v7
v0
e′6
e6
v6
e3
e′4
e′5
e5
v4
e4
v5
Figure 3.4: Wheel Graph W7
296
Consider the edge sets Ei = {ei , e′i+1 , ei+1 }, i = 2k − 1, . . . , 5 for k = 1, 2, 3, . . . that is,
297
i = 2(1) − 1 = 1 =⇒ E1 = {e1 , e′2 , e2 }
298
i = 2(2) − 1 = 3 =⇒ E2 = {e3 , e′4 , e4 }
299
i = 2(3) − 1 = 5 =⇒ E3 = {e5 , e′6 , e6 }
300
301
302
Ej = {e′i , e′i+2 , e′i+4 }, i = 6k − 3, . . . , 3 for k = 1, 2, 3, . . . that is,
i = 6(1) − 3 = 3 =⇒ E1 = {e′3 , e′5 , e′7 }
Ek = {e′1 , en }, that is, E1 = {e′1 , e7 }.
Main Results
24
303
Thus, the edge-induced subgraph ⟨Ei ⟩ forms 3 copies of claw S3 and ⟨Ej ⟩ forms a claw
304
S3 . Moreover, ⟨Ek ⟩ forms a path P3 . Hence, W7 decomposed into four copies of claw S3
305
and one copy of path P3 .
v2
e1
e2
v1
v3
e′2
v1
v3
v3
e′1
e3
v0
v0
e′4
v0
v7
v6
v4
v0
e5
e7
v0
v7
v7
e′5
e4
e′6
e6
e′7
e′3
v5
v5
v5
Figure 3.5: Decomposition of W7 into 4 copies of claw S3 and one copy of path P3
306
307
Case 3. If n ≡ 2 (mod 6), then n = 6d + 2 for d = 1, 2, 3, . . ..
For d = 1, then n = 8 since 8 ≡ 2 (mod 6).
the
wheel
graph
W8
with
Consider
309
{e1 , e2 , . . . , e8 , e′1 , e′2 , . . . , e′8 }. We show that W8 can be decomposed into [(n − 2d) − 1]S3
310
and P2 that is, W8 can decomposed into (n − 2d) − 1 = (6d + 2 − 2d) − 1 =
311
6(1) + 2 − 2(1) − 1 = 5 S3 and P2 .
v1
vertex
e1
e′2
set
v3
e′3
e′7
v7
edge
e2
e′1
e′8
e7
and
v2
e8
v8
set
{v0 , v1 , . . . , v8 }
308
v0
e′6
e′5
e3
e′4
e6
v4
e4
v6
e5
v5
Figure 3.6: Wheel Graph W8
312
Consider the edge sets Ei = {ei , e′i+1 , ei+1 }, i = 2k − 1, . . . , 7 for k = 1, 2, 3, . . . that is,
Main Results
25
313
i = 2(1) − 1 = 1 =⇒ E1 = {e1 , e′2 , e2 }
314
i = 2(2) − 1 = 3 =⇒ E2 = {e3 , e′4 , e4 }
315
i = 2(3) − 1 = 5 =⇒ E3 = {e5 , e′6 , e6 }
316
i = 2(4) − 1 = 7 =⇒ E4 = {e7 , e′8 , e8 }
317
Ej = {e′i , e′i+2 , e′i+4 }, i = 6k − 5, . . . , 1 for k = 1, 2, 3, . . . that is,
i = 6(1) − 5 = 1 =⇒ E1 = {e′1 , e′3 , e′5 }
318
319
Ek = {e′n−1 } that is, E1 = {e′7 }.
320
This means that the edge-induced subgraph ⟨Ei ⟩ forms 4 copies of claws S3 and ⟨Ej ⟩ forms
321
a claw S3 . Also, ⟨Ek ⟩ forms a path P2 . Therefore, W8 can be decomposed into 5 copies of
claws S3 and one copy of path P2 .
v1
v2
e1
v1
v1
e2
e8
v8
e′2
e′8
v0
v3
v3
v0
e7
e3
v0
v7
v7
e6
v0
e′4
e′6
v6
v4
e′1
v3
e′3
v0
e′7
v0
v7
e′5
e4
e5
v5
v5
v5
Figure 3.7: Decomposition of W8 into 5 copies of claw S3 and one copy of path P2
322
323
324
325
Case 4. If n ≡ 3 (mod 6) then n = 6d − 3 for d = 1, 2, 3, . . ..
For d = 1, then n = 3 since 3 ≡ 3 (mod 6) such that 6 | 3 − 3.
Consider the wheel graph W3 with vertex set {v0 , v1 , v2 , v3 } and edge set {e1 , e2 , e3 , e′1 , e′2 , e′3 }.
Main Results
326
26
We show that W3 can be decomposed into (n − 2d)S3 and C3 that is, W3 can be decomposed
into (n − 2d) = 6d − 3 − 2d = 6(1) − 3 − 2(1) = 1 S3 and C3 .
v2
e′2
e1
e2
e′1
v0
v1
e′3
e3
v3
Figure 3.8: Wheel Graph W3
327
328
Consider the edge set Ei = {ei , e′i+1 , ei+1 }, i = 2k − 1, . . . , 1 for k = 1, 2, 3, . . . that is,
i = 2(1) = 1 =⇒ E1 = {e1 , e′2 , e2 }
329
330
And Ek = {e′1 , e′n , en } that is, E1 = {e′1 , e′3 , e3 }
331
Therefore, the edge-induced subgraph ⟨Ei ⟩ forms a claw and ⟨Ek ⟩ forms a cycle C3 . Hence,
W3 decomposed into one copy of claw S3 and one copy of cycle C3 .
v2
e′2
e1
e2
v0
e′1
v0
e′3
v1
v3
v1
e3
v3
Figure 3.9: Decomposition of W3 into one copy of claw S3 and one copy of cycle C3
332
333
For d = 2, then n = 9 since 9 ≡ 3 (mod 6) such that 6 | 9 − 3.
the
wheel
graph
W9
with
vertex
set
{v0 , v1 , . . . , v9 }
334
Consider
and
edge
set
335
{e1 , e2 , . . . , e9 , e′1 , e′2 , . . . , e′9 }. We show that W9 can be decomposed into (n − 2d)S3 and C3
336
that is, W9 can be decomposed into (n − 2d) = 6d − 3 − 2d = 6(2) − 3 − 2(2) = 5 S3 and C3 .
Main Results
27
v2
e1
v1
e9
e′1
e′9
v9
e′8
e8
v3
e′2
e3
e′3
v0
e′7
v8
e2
v4
e′4
e4
e′5
e′6
v5
e7
e5
v7
e6
v6
Figure 3.10: Wheel Graph W9
337
Consider the edge sets Ei = {ei , e′i+1 , ei+1 }, i = 2k − 1, . . . , 7 for k = 1, 2, 3, . . . that is,
338
i = 2(1) − 1 = 1 =⇒ E1 = {e1 , e′2 , e2 }
339
i = 2(2) − 1 = 3 =⇒ E2 = {e3 , e′4 , e4 }
340
i = 2(3) − 1 = 5 =⇒ E3 = {e5 , e′6 , e6 }
341
i = 2(4) − 1 = 7 =⇒ E4 = {e7 , e′8 , e8 }
342
343
Ej = {e′i , e′i+2 , e′i+4 }, i = 6k − 3, . . . , 3 for k = 1, 2, 3, . . . that is,
i = 6(1) − 3 = 3 =⇒ E1 = {e′3 , e′5 , e′7 }
344
And Ek = {e′1 , e′n , en } that is, E1 = {e′1 , e′9 , e9 }.
345
Therefore, the edge-induced subgraph ⟨Ei ⟩ forms 4 copies of claw S3 and ⟨Ej ⟩ forms a
346
claw S3 Moreover, ⟨Ek ⟩ forms a cycle C3 . Hence, W9 decomposed into 5 copies of claw S3
347
and one copy of cycle C3 .
Main Results
28
v2
e1
v1
e2
v3
v3
e′2
v3
v1
e3
v9
v0
e′4
v0
v8
v0
e′7
e5
v7
e6
v5
v5
e7
v7
v0
e′5
v5
e′6
v9
e′9
e4
v0
e′8
e8
v4
v0
e′1
e9
e′3
v7
v6
Figure 3.11: Decomposition of W9 into 5 copies of claw S3 and one copy of cycle C3
348
349
Case 5. If n ≡ 4 (mod 6) then 6d − 2 for d = 1, 2, 3, . . ..
For d = 1, then n = 4 since 4 ≡ 4 (mod 6) such that 6 | 4 − 4.
with vertex set {v0 , v1 , v2 , v3 , v4 } and edge set
350
Consider the wheel graph W4
351
{e1 , e2 , e3 , e4 , e′1 , e′2 , e′3 , e′4 }. We show that W4 can be decomposed into (n − 2d)S3 and P3 .
352
That is, W4 can be decomposed into (n − 2d) = (6d − 2 − 2d) = 6(1) − 2 − 2(1) = 2 S3 and
353
P3 .
v1
v2
e1
e′1
e′2
e4
e2
e′4
v4
v0
e′3
e3
v3
Figure 3.12: Wheel Graph W4
354
Consider the edge set Ei = {ei , e′i+1 , ei+1 }, i = 2k − 1, . . . , 3 for k = 1, 2, 3, . . . that is,
355
i = 2(1) = 1 =⇒ E1 = {e1 , e′2 , e2 }
356
i = 2(2) = 3 =⇒ E2 = {e3 , e′4 , e4 }
Main Results
29
357
And Ek = {e′1 , e′n−1 } that is, E1 = {e′1 , e′3 }.
358
Thus, the edge-induced subgraph ⟨Ei ⟩ forms 2 copies of claw S3 and ⟨Ek ⟩ forms a path
P3 . Hence, W4 decomposed into a two copies of claw S3 and one copy of path P2 .
v1
v2
e1
v1
v1
e′1
e′2
v0
e4
e2
v0
v0
e′4
e′3
v3
v4
v3
e3
v3
Figure 3.13: Decomposition of W4 into 2 copies of claw S3 and one copy of path P3
359
360
For d = 2, then n = 10 since 10 ≡ 4 (mod 6) such that 6 | 10 − 4.
with vertex set {v0 , v1 , . . . , v10 } and edge set
361
Consider the wheel graph W10
362
{e1 , e2 , . . . , e10 , e′1 , e′2 , . . . , e′10 }. We show that W10 can be decomposed into (n − 2d)S3 and
363
P3 . That is, W10 can be decomposed into (n − 2d) = (6d − 2 − 2d) = 6(2) − 2 − 2(2) = 6 S3
364
and P3 .
v2
v1
e2
e3
e′1
e10
e′2
v0
e′9
e9
e′8
v9
e′7
e4
e′6
e5
e6
e7
v5
e′5
e8
v8
v4
e′3
e′4
e′10
v10
v3
e1
v6
v7
Figure 3.14: Wheel Graph W10
365
366
Consider the edge sets Ei = {ei , e′i+1 , ei+1 }, i = 2k − 1, . . . , 9 for k = 1, 2, 3, . . . that is,
i = 2(1) − 1 = 1 =⇒ E1 = {e1 , e′2 , e2 }
Main Results
30
367
i = 2(2) − 1 = 3 =⇒ E2 = {e3 , e′4 , e4 }
368
i = 2(3) − 1 = 5 =⇒ E3 = {e5 , e′6 , e6 }
369
i = 2(4) − 1 = 7 =⇒ E4 = {e7 , e′8 , e8 }
370
i = 2(5) − 1 = 9 =⇒ E5 = {e9 , e′10 , e10 }
371
Ej = {e′i , e′i+2 , e′i+4 }, i = 6k − 3, . . . , 3 for k = 1, 2, 3, . . . that is,
i = 6(1) − 3 = 3 =⇒ E1 = {e′3 , e′5 , e′7 }
372
373
And Ek = {e′1 , e′n−1 } that is, E1 = {e′1 , e′9 }.
374
Thus, the edge-induced subgraph ⟨Ei ⟩ forms 5 copies of claw S3 and ⟨Ej ⟩ forms a claw
375
S3 . Also, ⟨Ek ⟩ forms a path P3 . Hence W10 decomposed into six copies of claw S3 and one
copy of path P2 .
v2
e2
v3
v3
e1
v1
e3
e′2
v1
v1
e′10
v0
v0
v0
e′6
e5
e′8
v9
v9
v0
e′9
e′7
v9
e6
e8
v8
v5
e′5
v5
v0
e′1
v5
v0
v0
e9
e′3
e4
e′4
e10
v10
v3
v4
e7
v7
v6
v7
v7
Figure 3.15: Decomposition of W10 into 6 copies of claw S3 and one copy of path P3
376
377
378
379
Case 6. If n ≡ 5 (mod 6), then n = 6d − 1 for d = 1, 2, 3, . . ..
For d = 1, then n = 5 since 5 ≡ 5 (mod 6) such that 6 | 5 − 5.
Consider
the
wheel
graph
W5
with
vertex
set
{v0 , v1 , . . . , v5 }
and
edge
set
Main Results
31
380
{e1 , . . . , e5 , e′1 , . . . , e′5 }. We show that W5 can be decomposed into (n − 2d)S3 and P2 . That
381
is, W5 can decomposed into (n − 2d) = 6d − 1 − 2d = 6(1) − 1 − 2(1) = 3 S3 and P2 .
v2
e1
e′2
v1
e2
v3
e′1
e′3
v0
e5
e′5
v5
e3
e′4
e4
v4
Figure 3.16: Wheel Graph W5
382
Consider the edge sets Ei = {ei , e′i+1 , ei+1 }, i = 2k − 1, . . . , 3 for k = 1, 2, 3, . . . that is,
383
i = 2(1) − 1 = 1 =⇒ E1 = {e1 , e′2 , e2 }
384
i = 2(2) − 1 = 3 =⇒ E2 = {e3 , e′4 , e4 }
385
386
Ej = {e′i , e′i+2 , e′i+4 }, i = 6k − 5, . . . , 1 for k = 1, 2, 3, . . . that is,
i = 6(1) − 5 = 1 =⇒ E1 = {e′1 , e′3 , e′5 }
387
And Ek = {en } that is, E1 = {e5 }.
388
Therefore, the edge-induced subgraph ⟨Ei ⟩ forms 2 copies of claw S3 and ⟨Ej ⟩ forms a
389
claw S3 . Moreover, ⟨Ek ⟩ forms a path P2 . Hence, W5 decomposed into three copies of claw
390
S3 and one copy of path P2 .
391
For d = 2, then n = 11 since 11 ≡ 5 (mod 6) such that 6 | 11 − 5.
with vertex set {v0 , v1 , . . . , v11 } and edge set
392
Consider the wheel graph W11
393
{e1 , e2 , . . . , e11 , e′1 , e′2 , . . . , e′11 }. We show that W11 can be decomposed into (n − 2d)S3 and
Main Results
32
v2
e1
e2
e′2
v1
v3
v1
v3
v0
v3
e′1
v0
e′3
v0
e′4
v5
e4
e5
e′5
e3
v1
v5
v5
v4
Figure 3.17: Decomposition of W5 into 3 copies of claw S3 and one copy of path P2
394
P2 . That is, W11 can decomposed into (n − 2d) = 6d − 1 − 2d = 6(2) − 1 − 2(2) = 7 S3 and
395
P2 .
v3
e2
v2
v1
e1
e′1
e11
e′2
e10
v10
e′10
e′3
e4
v5
e′4
v0
e′9
e′8
e9
v9
v4
e5
e′5
e′11
v11
e3
e′7
e6
e7
e8
v6
e′6
v7
v8
Figure 3.18: Wheel Graph W11
396
Consider the edge sets Ei = {ei , e′i+1 , ei+1 }, i = 2k − 1, . . . , 9 for k = 1, 2, 3, . . . that is,
397
i = 2(1) − 1 = 1 =⇒ E1 = {e1 , e′2 , e2 }
398
i = 2(2) − 1 = 3 =⇒ E2 = {e3 , e′4 , e4 }
399
i = 2(3) − 1 = 5 =⇒ E3 = {e5 , e′6 , e6 }
400
i = 2(4) − 1 = 7 =⇒ E4 = {e7 , e′8 , e8 }
401
i = 2(5) − 1 = 9 =⇒ E5 = {e9 , e′10 , e10 }
Main Results
402
33
Ej = {e′i , e′i+2 , e′i+4 }, i = 6k − 5, . . . , 7 for k = 1, 2, 3, . . . that is,
403
i = 6(1) − 5 = 1 =⇒ E1 = {e′1 , e′3 , e′5 }
404
i = 6(2) − 5 = 7 =⇒ E2 = {e′7 , e′9 , e′11 }
405
And Ek = {en } that is, E1 = {e11 }.
406
Thus, the edge-induced subgraph ⟨Ei ⟩ forms 5 copies of claw S3 and ⟨Ej ⟩ forms 2 copies
407
of claw S3 . Moreover, ⟨Ek ⟩ forms a path P2 . Hence W11 decomposed into seven copies of
408
claw S3 and one copy of path P2 .
v3
v3 v3
e2
v2
v1
e3
v4
e1
e4
e′2
v5
e′4
v0
e10
e′10
v0
v6
e′6
e6
e′8
v10
e9
v9
v9
v11
e′9
v7
v8
e11
v0
v0
e′11
v7
e7
e8
v1
e′5
e5
v0
v5
e′1
v5
v0 v0
v11
e′3
v1
v11
e′7
v7
v9
Figure 3.19: Decomposition of W11 into 7 copies of claw S3 and one copy of path P2
409
Remark 3.3. “In the above theorem, Case 1 guarantees that there is a claw decomposition
410
for Wn .”
411
Proof. Consider the graphs in Figure 3.3.
412
edge-induced subgraph that forms S3 is isomorphic to each other. Thus, there is a claw
413
decomposition for Wn if n ≡ 0 (mod 6) such that n = 6d for d = 1, 2, 3, . . ..
By Definition 2.17, observe that each
414
Chapter 4
415
Summary and Recommendation
This chapter includes a summary of the results of this expository paper and
416
417
recommendations for further research.
418
4.1
Summary
419
We studied in detail the “decomposition of wheel graphs into claws, cycles and
420
paths” based on work of M. Subbulakshmi and I. Valliammmal entitled “Decomposition of
421
Wheel Graphs into Stars, Cycles and Paths” [10]. Also, we gave examples to illustrate the
422
following conditions for such a decomposition to exist:
“
D(Wn ) =




(n − 2d)S3 , d = 1, 2, 3, . . .
if n ≡ 0









[(n − 2d) − 1] S3 and P3 , d = 1, 2, 3, . . . if n ≡ 1







 [(n − 2d) − 1] S3 and P2 , d = 1, 2, 3, . . . if n ≡ 2



(n − 2d)S3 and C3 , d = 1, 2, 3, . . .









(n − 2d)S3 and P3 , d = 1, 2, 3, . . .







 (n − 2d)S3 and P2 , d = 1, 2, 3, . . .
423
424
425
(mod 6)
(mod 6)
(mod 6)
.
if n ≡ 3
(mod 6)
if n ≡ 4
(mod 6)
if n ≡ 5
(mod 6)
”
We discovered in Theorem 3.2 Case 1 guarantees that there is a “claw decomposition”
for Wn if n ≡ 0 (mod 6) such that n = 6d for d = 1, 2, 3, . . ..
34
Summary and Recommendation
426
427
428
4.2
35
Recommendation
A worthwhile direction for further investigation is to establish other possible
components of wheel graph’s decomposition.
List of References
429
430
[1] V.M. Abraham, S. Arumugam, and I. Sahul Hamid, Decomposition of graphs into paths
and cycles, Journal of Discrete Mathematics 721051 (2013), 6 pages.
431
432
[2] J.A Bondy and U.S.R. Murty, Graph theory with applications, The Macmillan Press,
New York, 1976.
433
434
[3] Susanna S. Epp, Discrete mathematics with applications 4th edition, Cengage Learning,
2011.
435
436
[4] L.T. Cherin Monish Femila and S. Asha, Hamiltonian decomposition of wheel related
437
graphs, International Journal of Scientific Research and Review 7(11) (2018), 338–345.
438
[5] F. Harary, Graph theory, MA: Addison-Wesley, 1969.
439
[6] V. Rajeswari and K. Thiagarajan, Graceful labeling of wheel graph and middle graph
440
of wheel graph under ibede and sibede approach, National Conference on Mathematical
441
Techniques and its Applications (NCMTA-18).
442
[7] Tay-Woei Shyu, Decompositions of complete graphs into paths and cycles, Ars
Combinatoria 97 (2010), 257–270.
443
444
[8]
, Decompositions of complete graphs into paths and stars, Discrete Mathematics
310 (2010), 2164–2169.
445
446
[9] M. Subbulaksmi and I. Valliammal, Decomposition of generalized petersen graphs into
447
paths and cycles, International Journal of Mathematics Trends and Technology (SSRG-
448
IJMTT)-Special Issue NCPAM NCPAMP109 (2019), 62–67.
449
450
[10]
, Decomposition of wheel graphs into stars, cycles and paths, Malaya Journal of
Matematik Vol. 9 No. 1 (2021), 456–460.
36