Ubiquity of graphs with nowhere-linear end structure

Abstract

A graph G G is said to be less than or similar to-ubiquitous, where less than or similar to is the minor relation between graphs, if whenever Gamma is a graph with Gamma for all n is an element of N, then one also has aleph(0)G less than or similar to Gamma where alpha G is the disjoint union of alpha many copies of G. A well-known conjecture of Andreae is that every locally finite connected graph is less than or similar to-ubiquitous. In this paper we give a sufficient condition on the structure of the ends of a graph G which implies that G is less than or similar to-ubiquitous. In particular this implies that the full-grid is less than or similar to-ubiquitous.