Malaya Journal of Matematik, Vol. 9, No. 1, 456-460, 2021

https://doi.org/10.26637/MJM0901/0076

Decomposition of wheel graphs into stars, cycles and paths

M. Subbulakshmi1 and I. Valliammal2*

Abstract Let G = (V,E) be a finite graph with n vertices. The Wn is a graph with set {v0,v1,v2,...,vn} and edge-set consisting of all edges of the form vivi+1 and v0vi where 1 ≤ i ≤ n, the subscripts being reduced modulo n. Wheel graph of (n + 1) vertices denoted by Wn. Decomposition of Wheel graph denoted by D(Wn). A with 3 edges is called a claw S3. In this paper, we show that any Wheel graph can be decomposed into following ways.  (n − 2d)S , d = 1,2,3,... if n ≡ 0 (mod 6)  3  [(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) D(Wn) = . (n − 2d)S and C , d = 1,2,3,... if n ≡ 3 (mod 6)  3 3  (n − 2d)S3 and P3, d = 1,2,3... if n ≡ 4 (mod 6)  (n − 2d)S3 and P2, d = 1,2,3,... if n ≡ 5 (mod 6) Keywords Wheel Graph, Decomposition, claw. AMS Subject Classification 05C70. 1Department of Mathematics, G.V.N. College, Kovilpatti, Thoothukudi-628502, Tamil Nadu, India. 2Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli-627012, Tamil Nadu, India. *Corresponding author: [email protected]; [email protected] Article History: Received 01 November 2020; Accepted 30 January 2021 c 2021 MJM.

Contents to all vertices of a cycle. A Wheel graph with n vertices can also be defined as the 1-skeleton of an 1 Introduction...... 456 n-gonal . Wheel graph of (n + 1) vertices denoted by 2 Basic Definitions...... 456 Wn. Number of edges of Wheel graph with (n + 1) vertices is 2n. A star Sn is the complete bipartite graph K1,n; A 3 Examples of Wheel graphs...... 457 with one internal vertex and n edges. 4 Decomposition of Wheel Graph Wn ...... 457 All graphs considered here are finite and undirected with- References...... 460 out loops, unless otherwise noted. For the standard graph- theoretic terminology the reader is referred to [8] and to study about the decomposition of graphs into paths, stars and cycles 1. Introduction is referred to [1], [2], [3] and [4]. is proved to be tremendously useful in mod- As usual Cn denotes the cycle of length n, Pn+1 denotes eling the essential features of system with finite components. the path of length n, S3 denotes the claw and D(Wn)] denotes Graphical models are used to represent telephone network, decomposition of Wheel graph. railway network, communication problems, traffic network etc. A graph is a convenient way of representing informa- tion involving relationship between objects. The objects are 2. Basic Definitions represented by vertices and the relations by edges [5]. In this section, we see some basic definitions of graph A Wheel graph is a graph formed by connecting a single decomposition and Wheel graph. Decomposition of wheel graphs into stars, cycles and paths — 457/460

Let L = {H1,H2,...,Hr} be a family of subgraphs of 3. Examples of Wheel graphs G. An L − decomposition of G is an edge- disjoint decom- In this section, we see some examples of Wheel Graphs. position of G into positive integer αi copies of Hi, where i ∈ W and W are represented in Figure 1.1 and Figure {1,2,...,r}. Furthermore, if each Hi (i ∈ {1,2,...,r}) is iso- 6 8 morphic to a graph H, we say that G has an H −decomposition. 1.2 respectively. r It is easily seen that ∑i=1 αi e(Hi) = e(G) is one of the obvi- ous necessary conditions for the existence of a {H1,H2,...,Hr}− 4. Decomposition of Wheel Graph Wn decomposition of G. For convenience, we call the equation, r In this section, we characterize the theorem of decomposi- ∑i=1 αi e(Hi) = e(G), a necessary sum condition. tion of Wheel graph Wn into claws, cycles and paths.

Let G = (V,E) be a finite graph with n vertices. The Theorem 4.1. Any Wheel Wn,n ≥ 3 can be decomposed into Wheel graph Wn is a graph with vertex set {v0,v1,v2,...,vn} Cn and Sn. and edge-set consisting of all edges of the form vivi+ and 1 Proof. Proof is immediate from the definition of the Wheel v vi where 1 ≤ i ≤ n, the subscripts being reduced modulo n. 0 graph. Wheel graph of (n + 1) vertices denoted by Wn.

Theorem 4.2. Any Wheel graph Wn,n ≥ 3 can be decomposed Obviously Wheel graph Wn has every vertex of degree 3 into following ways. except the internal vertex. Internal vertex has degree n.

Figure 1. Examples of Wheel Graphs

0 where ei and ei, 1 ≤ i ≤ n are edges in outer and inner cycles. (n − 2d)S , d = 1,2,3,...  3   if n ≡ 0 (mod 6) Case 1. n ≡ 0 (mod 6).  [(n − 2d) − 1]S and P , d = 1,2,3... Then n = 6d,d = 1,2,3,....  3 3  To Prove: Wn decomposed into (n − 2d)S3. That is to prove  if n ≡ 1 (mod 6)  that Wn decomposed into 4dS3, since n − 2d = 6d − 2d = 4d. [(n − 2d) − 1]S3 and P2, d = 1,2,3,... n 0 o  Let Ei = ei,e ,ei+1 where i = 1,3,5,...,  if n ≡ 2 (mod 6) i+1 D(Wn) = n 0 0 0 o (n − 2d)S and C , d = 1,2,3,... n − 1 and E j = ei,ei+2,ei+4 where i = 1,7,13,...,n − 5.  3 3  The edge induced subgraph < E > forms 3d claws and the  if n ≡ 3 (mod 6) i  edge induced subgraph < E j > forms d claws. Hence Wn (n − 2d)S3 and P3, d = 1,2,3...  decomposed into 3d + d = 4d claws.   if n ≡ 4 (mod 6)  (n − 2d)S3 and P2, d = 1,2,3,...  Case 2. n ≡ 1 (mod 6).  if n ≡ 5 (mod 6) Then n = 6d + 1,d = 1,2,3,.... Proof. Let V(Wn) = {v0,v1,v2,...,vn} and To Prove: Wn decomposed into [(n − 2d) − 1] S P W dS n 0 0 0 o 3 and 3. That is to prove that n decomposed into 4 3 and E(Wn) = e1,e2,...,en,e1,e2,...,en P3, since (n − 2d) − 1 = (6d + 1 − 2d) − 1 = 4d.

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n 0 o induced subgraph < E > forms d claws and the edge induced Let Ei = ei,ei+1,ei+1 where i = 1,3,5,..., j n 0 0 0 o subgraph < Ek > forms a path P3 of length 2. Hence Wn de- n − 2, E = e ,e ,e where i = 3,9,15, j i i+2 i+4 composed into 3d + d = 4d claws and a path P3. n 0 o ...,n − 4 and Ek = e ,en . 1 Illustration: Decomposition of Wheel graph on case 1 The edge induced subgraph < E > forms 3d claws, the edge i and case 2 explained through the following Figure-2.

Figure 2.

n 0 0 0 o Figures 2. represents decomposition of W into 4 claws 6 n − 2, E j = ei,ei+2,ei+4 where i = 3,9,15, n 0 0 o and decomposition of W7 into 4 claws and a path P3 respec- ...,n − 6 and Ek = e1,en,en . tively. The edge induced subgraph < Ei > forms 3d − 2 claws, the All edges of the claws and path differentiated in the above edge induced subgraph < E j > forms d − 1 claws and the Figure-2. edge induced subgraph < Ek > forms one cycle C3 of length 3. Hence Wn decomposed into 3d − 2 + d − 1 = 4d − 3 claws Case 3. n ≡ 2 (mod 6). and a cycle C3. Then n = 6d + 2,d = 1,2,3,.... To Prove: Wn decomposed into [(n − 2d) − 1] Illustration: Decomposition of Wheel graph on case 3 and case 4 explained through the following Figure-3. S3 and P2. That is to prove that Wn decomposed into (4d + The above two figures represents decomposition of W 1)S3 and P2, since (n−2d)−1 = (6d +2−2d)−1 = 4d +1. 8 n 0 o into 5 claws and a path P2 and decomposition of W9 into 5 Let Ei = ei,ei+1,ei+1 where i = 1,3,5,..., claws and a cycle C3 respectively. n 0 0 0 o n − 1, E j = ei,ei+2,ei+4 where i = 1,7,13, All edges of the claws and path differentiated in the above n 0 o Figure-3. 19,...,n − 7 and Ek = en−1 . The edge induced subgraph < Ei > forms 3d + 1 claws, the Case 5. edge induced subgraph < E j > forms d claws and the edge n ≡ 4 (mod 6). induced subgraph < Ek > forms a path P2 of length 1. Hence Then n = 6d − 2,d = 1,2,3,.... Wn decomposed into 3d +1+d = 4d +1 claws and a path P2. To Prove: Wn decomposed into (n − 2d)S3 and P3. That is to prove that Wn decomposed into (4d − 2)S3 and P3, since Case 4. n ≡ 3 (mod 6). n − 2d = 6d − 2 − 2d = 4d − 2. n 0 o Then n = 6d − 3,d = 1,2,3,.... Let Ei = ei,ei+1,ei+1 where i = 1,3,5,..., To Prove: Wn decomposed into (n − 2d)S3 and C3. That is n 0 0 0 o n − 1, E j = ei,ei+2,ei+4 where i = 3,9,15, to prove that Wn decomposed into (4d − 3)S3 and C3, since n 0 0 o n − 2d = 6d − 3 − 2d = 4d − 3. ...,n − 7 and Ek = e ,e . n 0 o 1 n−1 Let Ei = ei,ei+1,ei+1 where i = 1,3,5,...,

458 Decomposition of wheel graphs into stars, cycles and paths — 459/460

Figure 3.

n 0 o The edge induced subgraph < E > forms 3d − 1 claws, i Let Ei = ei,ei+1,ei+1 where i = 1,3,5,..., the edge induced subgraph < E j > forms d − 1 claws and the n 0 0 0 o n − 2, E = e ,e ,e where i = 1,7,13, edge induced subgraph < Ek > forms a path P3 of length 2. j i i+2 i+4 Hence Wn decomposed into 3d − 1 + d − 1 = 4d − 2 claws ...,n − 4 and Ek = {en}. and a path P3. The edge induced subgraph < Ei > forms 3d − 1 claws, the Case 6. edge induced subgraph < E j > forms d claws and the edge n ≡ 5 (mod 6). induced subgraph < Ek > forms a path P2 of length 1. Hence Then n = 6d − 1,d = 1,2,3,.... Wn decomposed into 3d −1+d = 4d −1 claws and a path P2. To Prove: Wn decomposed into (n − 2d)S3 and P2. That is to prove that Wn decomposed into (4d − 1)S3 and P2, since n − 2d = 6d − 1 − 2d = 4d − 1. Illustration: Decomposition of Wheel graph on case 5 and case 6 explained through the following Figure-4.

Figure 4.

Figures 4 represents decomposition of W10 into 6 claws All edges of the claws and path differentiated in the above and a path P3 and decomposition of W11 into 7 claws and a Figure-4. path P2 respectively.

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Remark 4.3. In the above theorem Case 1 guarantees that there is a claw decomposition for Wn.

References [1] Tay-Woei Shyu, Decompositions of Complete Graphs into Paths and Cycles, Ars Combinatoria, 97(2010), 257- 270. [2] Tay-Woei Shyu, Decompositions of Complete Graphs into Paths and Stars, Discrete Mathematics, 310(2010), 2164-2169. [3] S. Arumugam, I. Sahul Hamid and V.M. Abraham, De- composition of graphs into Paths and Cycles, Journal of Discrete Mathematics, 721051(2013), 6 pages. [4] Tay-Woei Shyu, Decompositions of Complete Graphs into Cycles and Stars, Graphs and Combinatorics, 29(2013), 301-313. [5] L.T. Cherin Monish Femila and S. Asha, Hamiltonian Decomposition of Wheel Related Graphs, International Journal of Scientific Research and Review, 7(11)(2018), 338-345. [6] Sujatha. M, Sasikala. R, Mathivanan. R, A study on De- composition of graphs, Emerging Trends in Pure and Applied Mathematics (ETPAM-2018)- 2018. [7] V. Rajeswari and K. Thiagarajan, of Wheel Graph and Middle Graph of Wheel Graph under IBEDE and SIBEDE Approach, National Conference on Mathematical Techniques and its Applications (NCMTA- 18). [8] J. A. Bondy, U.S.R. Murty, Graph theory with applica- tions, The Macmillan Press Ltd, New Yark, 1976. [9] West,D, Introduction to Graph Theory, third edn. Prentice Hall, Saddle River (2007). [10] Michele Conforti, Gerard Cornuejols and M.R. Rao, De- composition of wheel-and-parachute-free balanced bi- partite graphs, Discrete Applied Mathematics, 62(1995), 103-117. [11] M. Subbulakshmi, I. Valliammal, Decomposition of Gen- eralized Petersen Graphs into Paths and Cycles, Interna- tional Journal of Mathematics Trends and Technology (SSRG- IJMTT)- Special Issue NCPAM, 2019, NCPAM- P109, 62-67.

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