Decomposition of complete graphs into paths and stars

Let P k + 1 denote a path of length k and let S k + 1 denote a star with k edges. As usual K n denotes the complete graph on n vertices. In this paper we investigate the decomposition of K n into paths and stars, and prove the following results. m A. Let p and q be nonnegative integers and let n be a positive integer. There exists a decomposition of K n into p copies of P 4 and q copies of S 4 if and only if n ≥ 6 and 3 ( p + q ) = n 2 . m B. Let p and q be nonnegative integers, let n and k be positive integers such that n ≥ 4 k and k ( p + q ) = n 2 , and let one of the following conditions hold: (1)  even and  p ≥ k 2 ,  odd and  p ≥ k . there exists a decomposition of K n into p copies of P k + 1 and q copies of S k + 1 .