A. Frank (Problem session of the Fifth British Combinatorial Conference, Aberdeen, Scotland, 1975) conjectured that if G = (V, E) is a connected graph with all valencies ≥k and a1,...,ak ≥ 2 are integers with Σ ai = |V |, then V may be decomposed into subsets A1,...,Ak so that |Ai | = ai and the subgraph spanned by Ai in G has no isolated vertices (i = 1,...,k). The case k = 2 is proved in Maurer (J. Combin. Theory Ser. B 27 (1979), 294-319) along with some extensions. The conjecture for k = 3 and a result stronger than Maurer's extension for k = 2 are proved. A related characterization of a k-connected graph is also included in the paper, and a proof of the conjecture for the case a1 = a2 = ... = ak-1 = 2.