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