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. Theorem 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)=n2. Theorem B. Let p and q be nonnegative integers, let n and k be positive integers such that n>=4k and k(p+q)=n2, and let one of the following conditions hold: (1)k is even and p>=k2, (2)k is odd and p>=k. Then there exists a decomposition of K"n into p copies of P"k"+"1 and q copies of S"k"+"1.