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Abstract. In this paper, it is shown that a necessary and sufficient condition for the existence of a Ps-factorization of K~ is (i) mn =-0(mod 3) and (ii) (m -1)n -= 0(mod 4).Let G and H be graphs. A spanning subgraph F of G is called an H-factor if and only if each component ofF is isomorphic to H. If G is expressible as an edge-disjoint sum of H-factors, then this sum is called an H-factorization of G.Let P3 be a path on three vertices and K~ be a complete multipartite graph with m partite sets of n vertices each (m > 2, n > 1). In this paper, we shall prove the following theorem: and n -0(mod 6), and (VI) n = 0(mod 12). Lemma 2. Let G, H and I be graphs. If G has an H-factorization and H has an I-factorization, then G has an I-factorization.Proof. Let E(G) = U,'.=I E(F/) be an H-factorization of G. Let H~ ° (1 < j < t) be the components of Fi. And let E(H) °) = U~k=l E(Itk ida) be an I-factorization of H~ °. Then E(G) = ~= 1 U~,=I E(U~=I Ik ~iJ)) is an I-factorization of G. Proof. We can obtain K~ from K~, by replacing each edge (i.e. each subgraph K1,1) by Ks, s. Using a P3-factorization of K~, we can see that K~ has a Ks, 2s-factorization. Since Ks, 2s has a P3-factorization (see I-4] J, K~," has a P3-factorization.