Conditions for families of disjoint k-connected subgraphs in a graph

In [Many disjoint dense subgraphs versus large k-connected subgraphs in large graphs with given edge density, Discrete Math. 309 (2009), 997–1000.], Böhme and Kostochka showed that every large enough graph with sufficient edge density contains either a kconnected 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 2.5ksn ǫ edges contains a family of s disjoint k-connected subgraphs each of order at most ǫn. We also show for k ≥ 2, the vertices of a graph with minimum degree at least 2k√n can be partitioned into k-connected subgraphs. The degree condition in the latter result is asymptotically best possible as a function of n.