Decomposition of complete graphs into paths and stars

AbstractLet Pk+1 denote a path of length k and let Sk+1 denote a star with k edges. As usual Kn denotes the complete graph on n vertices. In this paper we investigate the decomposition of Kn 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 Kn into p copies of P4 and q copies of S4 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 Kn into p copies of Pk+1 and q copies of Sk+1