Abstract

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.

Author Keywords: Graph partitions; Minimum degree; Connectivity; Topological minors

Article Outline

1. Introduction

2. Proof of Theorems 1 and 2Theorems 1 and 2

3. Proof of Theorem 5

4. Partitions with constraints on the average degree or the chromatic number

References