Graph Decompositions without Isolated Vertices

We prove the following conjecture of A. Frank (Fifth British Combinatorial Conference, Aberdeen, Scotland, 1975): Let G be a connected simple graph of order n, and n = n1 + ··· +nk be a partition of n with ni ≥ 2. Suppose that the minimum degree of G is at least k. Then the vertex set V(G) can be decomposed into disjoint subsets V1, ..., Vk so that |Vi| = ni and the subgraph induced by Vi contains no isolated vertices for all i, 1 ≤ i ≤ k.