Dense on-line arbitrarily partitionable graphs

Abstract

A graph G of order n is called arbitrarily partitionable (AP, for short) if, for every sequence (n1,…,nk) of positive integers with n1+…+nk=n, there exists a partition (V1,…,Vk) of the vertex set V(G) such that Vi induces a connected subgraph of order ni, for i=1,…,k. In this paper we consider the on-line version of this notion, defined in a natural way.We prove that if G is a connected graph such with the independence number at most ⌈n2⌉ and the degree sum of any pair of non-adjacent vertices is at least n−3, then G is on-line arbitrarily partitionable except for two graphs of small orders. We also prove that if G is a connected graph of order n and size ‖G‖>n−32+6, then G is on-line AP unless n is even and G is a spanning subgraph of a unique exceptional graph. These two results imply that dense AP graphs satisfying one of the above two assumptions are also on-line AP. This is in contrast to sparse graphs where only few AP graphs are also on-line AP.