Partition conditions and vertex-connectivity of graphs

Abstract

It was proved ([5], [6]) that ifG is ann-vertex-connected graph then for any vertex sequencev 1, ...,v n ≠V(G) and for any sequence of positive integersk 1, ...,k n such thatk 1+...+k n =|V(G)|, there exists ann-partition ofV(G) such that this partition separates the verticesv 1, ...,v(n), and the class of the partition containingv i induces a connected subgraph consisting ofk i vertices, fori=1, 2, ...,n. Now fix the integersk 1, ...,k n . In this paper we study what can we say about the vertex-connectivity ofG if there exists such a partition ofV(G) for any sequence of verticesv 1, ...,v n ≠V(G). We find some interesting cases when the existence of such partitions implies then-vertex-connectivity ofG, in the other cases we give sharp lower bounds for the vertex-connectivity ofG.