Partitions of graphs with high minimum degree or connectivity

We prove that there exists a function f ðcÞ such that the vertex set of every f ðcÞ-connected graph G can be partitioned into sets S and T such that each vertex in S has at least c neighbours in T and both G1⁄2S and G1⁄2T are c-connected. This implies that there exists a function gðc;HÞ such that every gðc;HÞ-connected graph contains a subdivision TH of H so that G VðTHÞ is c-connected. We also prove an analogue with connectivity replaced by minimum degree. Furthermore, we show that there exists a function hðcÞ such that the vertex set of every graph G of minimum degree at least hðcÞ can be partitioned into sets S and T such that both G1⁄2S and G1⁄2T have minimum degree at least c and the bipartite subgraph between S and T has average degree at least c: r 2003 Elsevier Science (USA). All rights reserved.