DECOMPOSITION OF GENERALIZED PETERSEN GRAPHS INTO CLAWS, CYCLES AND PATHS

Let G = (V,E) be a finite graph with n vertices. Let n and k be positive integers where n ≥ 3 and 1 ≤ k < 2 . The Generalized Petersen Graph GP(n,k) is a graph with vertex set {u0,u1,u2, ...,un−1, v0,v1,v2, ...,vn−1} and edge-set consisting of all edges of the form uiui+1,uivi and vivi+k where 0 ≤ i ≤ n− 1, the subscripts being reduced modulo n. Obviously GP(n,k) is always a cubic graph and GP(5,2) is the well-known Petersen graph. In this paper, we show that GP(n,1),n ≥ 3 can be decomposed into n copies of S3 if n is even and P4 and (n−1) copies of S3 if n is odd. Also, we show that GP(n,2),n ≥ 5 can be decomposed into 2 copies of S3, 2 copies of C n 2 and 2 copies of P2 if n is even and Cn, P4 , ⌊ n 2 ⌋ copies of S3 and (⌊ n 2 ⌋ −1 ) copies of P2 if n is odd. GP(n,2),n≥ 5 and n = 3d,d = 2,3, ... can be decomposed into 2d copies of S3 and d copies of P4. GP(n,2),n≥ 5 and n = 4d,d = 2,3, ... can be decomposed into 3d copies of S3 and d copies of P4. GP(n,3),n ≥ 8 can be decomposed into n copies of S3 if n is even and P6, P2 and (n−2) copies of S3 if n is odd.


INTRODUCTION
The origin of graph decomposition is from the combinatorial problems most of which emerged in the 19th century. The first one was the prize question for the year in the Lady's and Gentlemen's Diary of 1844 stated by W.S.B.Woolhouse: Determine the number of combinations that can be made out of n symbols, p symbols in each such that no combination of q symbols which may appear in any one of them may appear in any other. When every q symbols appear in exactly one of the p subsets, such a configuration is known as a Steiner system. When q = 2 and p = 3, the configuration is known as Steiner Triple system. In 1847, T.P.Kirkman settled the existence question for the Steiner Triple system. It is equivalent to the decomposition of K n into triangles.
The other best problems are Kirkman's problem of 15 strolling school girls, Dudeney problem of 9 handcuffed prisoners, Euler's problem of 36 army officers, Kirkman's problem of knights, Lucas' dance around problem and the four color problem. However, the earliest works in the direction are not explicitly related to graph decompositions. The paper dealing directly with graph decomposition appeared after the turn of the 19th century. Since then, the interest in graph decomposition has been on increase and a real up surge is witnessed after 1950. Now a days, graph decomposition rank among the most prominent areas of graph theory and combinatorics. Many types of decomposition have been well studied in the literature. There are lot of applications of decomposition of graph which include group testings, DNA library screening, scheduling problems, sharing scheme and synchronous optical networks.
All graphs considered here are finite and undirected, 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 is referred to [1], [2], [3] and [4].
As usual C n denotes the cycle of length n, P n+1 denotes the path of length n and S 3 denotes the claw.

BASIC DEFINITIONS AND EXAMPLES OF GENERALIZED PETERSEN GRAPHS
In this section, we see some basic definitions of graph decomposition, Generalized Petersen  Let n and k be positive integers, n ≥ 3 and 1 ≤ k < n 2 . The Generalized Petersen Graph GP(n, k) is a graph with vertex set {u 0 , u 1 , u 2 , ..., u n−1 , v 0 , v 1 , v 2 , ..., v n−1 } and edge-set consisting of all egdes of the form u i u i+1 , u i v i and v i v i+k where 0 ≤ i ≤ n − 1, the subscripts being reduced modulo n.
Obviously GP(n, k) is always a cubic graph and GP(5, 2) is the well-known Petersen graph.
Thus GP(5, 2) is the Petersen graph and is represented in Figure 1(d).
Then each edge induced subgraph < E i > and < E j > forms d disjoint claws and the edge induced subgraph < E k > forms a path of length 4 (i.e, P 4 ). Therefore GP(n, 1) can be decomposed into d + d = 2d = n − 1 disjoint claws and P 4 . Hence GP(n, 1) can be decomposed into n − 1 disjoint claws and P 4 .

Illustration:
The above theorem can be explained through the following Figure 2.    GP(n, 2), n ≥ 5 can be decomposed into n 2 copies of S 3 , 2 copies of C n 2 and n 2 copies of P 2 if n is even and C n , P 4 , n 2 copies of S 3 and n 2 − 1 copies of P 2 if n is odd. Then, the edge induced subgraph < E i > forms d = n 2 disjoint claws, each edge induced subgraph < E j > and < E k > forms two disjoint cycles of length d = n 2 and the edge induced subgraph < E l > forms d = n 2 disjoint path of length one (i.e, P 2 ). Hence GP(n, 2) can be decomposed into n 2 S 3 , 2C n 2 and n 2 P 2 . Case 2. If n is odd. That is n = 2d − 1, d = 3, 4, .... Then, the edge induced subgraph < E i > forms n 2 disjoint claws, the edge induced subgraph < E j > forms a cycle of length n, the edge induced subgraph < E k > forms a path of length three (i.e, P 4 ) and the edge induced subgraph < E l > forms ( n 2 − 1) disjoint path of length one (i.e, P 2 ). Hence GP(n, 2) can be decomposed into n 2 claws, C n , P 4 and ( n 2 − 1)P 2 . Then each edge induced subgraph < E i > and < E j > forms d disjoint claws and the edge induced subgraph < E k > forms d disjoint paths of length 3 (i.e, P 4 ). Hence GP(n, 2), (n ≥ 5, n = 3d) can be decomposed into 2d claws and dP 4 . Then, the edge induced subgraph < E i > forms d disjoint claws, the edge induced subgraph < E j > forms 2d disjoint claws and the edge induced subgraph < E k > forms d disjoint path of length three (i.e, P 4 ). Hence GP(n, 2), (n ≥ 5, n = 4d) can be decomposed into 3d claws and dP 4 .

DECOMPOSITION OF GENERALIZED PETERSEN GRAPH GP(n, 3) INTO CLAWS AND PATHS
In this section, we characterize the theorem of decomposition of Generalized Petersen Graph GP(n, 3) into claws and paths.
Then each edge induced subgraph < E i > and < E j > forms n 2 disjoint claws. Hence GP(n, 3), n ≥ 8 can be decomposed into nS 3 if n is even.
Case 2. If n is odd and n ≥ 8.
Then, the edge induced subgraph < E i > forms n 2 disjoint claws, the edge induced subgraph < E j > forms n 2 − 1 disjoint claws, the edge induced subgraph < E k > forms a path of length five (i.e, P 6 ) and the edge induced subgraph < E l > forms a path of length one (i.e, P 2 ). Hence GP(n, 3), n ≥ 8 can be decomposed into and (n − 2) disjoint claws, P 6 and P 2 if n is odd.

Illustration:
The above theorem can be explained through the following Figure 3.
All edges of the claws and path differentiated in Figure 3.
Then the edge induced subgraph < E i > forms n 2 = 3 disjoint claws, the edge induced subgraph < E j > forms n 2 − 1 = 2 disjoint claws, the edge induced subgraph < E k > forms a cycle of length five (i.e, C 5 ) and the edge induced subgraph < E l > forms a path of length one (i.e, P 2 ). Hence GP(7, 3) can be decomposed into 5S 3 ,C 5 and P 2 .
Illustration: The above remark-5.2 can be explained through the following Figure 4. The above figure represents decomposition of GP(7, 3) into 5 claws, a cycle C 5 and a path P 2 respectively.
All edges of the claws, cycle and path differentiated in the above Figure 4.