Conditions for families of disjoint -connected subgraphs in a graph

In [T. Böhme A. Kostochka, Many disjoint dense subgraphs versus large k -connected subgraphs in large graphs with given edge density, Discrete Math. 309 (4) (2009) 997–1000.], Böhme and Kostochka showed that every large enough graph with sufficient edge density contains either a k -connected subgraph of order at least r or a family of r vertex-disjoint k -connected subgraphs. Motivated by this, in this note we explore the latter conclusion of their work and give conditions that ensure a graph G contains a family of vertex-disjoint k -connected subgraphs. In particular, we show that a graph of order n with at least 223 k s n 120 ϵ edges contains a family of s disjoint k -connected subgraphs each of order at most ϵ n . We also show for k ≥ 2 , the vertices of a graph with minimum degree at least 2 k n can be partitioned into k -connected subgraphs. The degree condition in the latter result is asymptotically the best possible as a function of n .