Decomposition of generalized Petersen graphs into claws, cycles and paths

Abstract

Let G = (V,E) be a finite graph with n vertices. Let n and k be positive integers where 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 edges of the form u i u i+1 ,uivi 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. In this paper, we show that GP(n,1), n ≥ 3 can be decomposed into n copies of S 3 if n is even and P 4 and (n−1) copies of S 3 if n is odd. Also, we show that GP(n,2), n ≥ 5 can be decomposed into n/2 copies of S3, 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. 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 S 3 and d copies of P 4 . GP(n,3), n ≥ 8 can be decomposed into n copies of S 3 if n is even and P 6 , P 2 and (n−2) copies of S 3 if n is odd.