Decomposition of complete graphs into paths of length three and triangles

Let P_k+1 denote a path of length k and let C _k denote a cycle of length k. A triangle is a cycle of length three. As usual K_n denotes the complete graph on n vertices. It is shown that for all nonnegative integers p and q and for all positive integers n, K_n can be decomposed into p copies of P₄ and q copies of C₃ if and only if 3(p + q) = e(K_n), p ‡1 if n is odd, and p ≥ n/2 1/2 n is even.