Graph decompositions through prescribed vertices without isolates

Let G be a graph of order n, and let n = P k i=1 a i be a partition of n with a i 2. Let v 1 ; : : : ; v k be given distinct vertices of G. Suppose that the minimum degree of G is at least 3k. In this paper, we prove that there exists a decomposition of the vertex set V (G) = S k i=1 A i such that jA i j = a i , v i 2 A i , and the subgraph induced by A i contains no isolated vertices for all i, 1 i k.