Graph decompositions without isolated vertices III

Let G be a graph of order n, and n = Σ^k_i=1 a_i be a partition of n with a_i ≥ 2. In this article we show that if the minimum degree of G is at least 3k−2, then for any distinct k vertices v₁,…, v_k of G, the vertex set V(G) can be decomposed into k disjoint subsets A₁,…, A_k so that |A_i| = a_i,v_iisA_i is an element of A_i and “the subgraph induced by A_i contains no isolated vertices” for all i, 1 ≥ i ≥ k. Here, the bound on the minimum degree is sharp. © 1997 John Wiley & Sons, Inc.