Partitions of graphs with high minimum degree or connectivity

We prove that there exists a function f(l) such that the vertex set of every f(l)-connected graph G can be partitioned into sets S and T such that each vertex in S has at least l neighbours in T and both G[S] and G[T] are l-connected. This implies that there exists a function g(l, H) such that every g(l, H)-connected graph contains a subdivision TH of H so that G - V(TH) is l-connected. We also prove an analogue with connectivity replaced by minimum degree. Furthermore, we show that there exists a function h(l) such that the vertex set of every graph G of minimum degree at least h(l) can be partitioned into sets S and T such that both G[S] and G[T] have minimum degree at least l and the bipartite subgraph between S and T has average degree at least l.