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 P₄ and q copies of S₄ if and only if n ≥ 6 and 3 (p + q) = fenced(frac(n, 2)). Theorem B. Let p and q be nonnegative integers, let n and k be positive integers such that n ≥ 4 k and k (p + q) = fenced(frac(n, 2)), and let one of the following conditions hold: (1)k is even and p ≥ frac(k, 2),(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}.