Graph decompositions without isolates

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,…,a_k ≥ 2 are integers with Σ a_i = |V |, then V may be decomposed into subsets A₁,…,A_k so that |A_i | = a_i and the subgraph spanned by A_i in G has no isolated vertices (i = 1,…,k). The case k = 2 is proved in Maurer (J. Combin. Theory Ser. B27 (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 = … = a_k?1 = 2.