Universal graphs for the topological minor relation

Abstract

A subgraph‐universal graph/a topological minor‐universal graph in a class of graphs is a graph in which contains every graph in as a subgraph/topological minor. We prove that the class of all countable planar graphs does not contain a topological minor‐universal graph. This answers a question of Diestel and Kühn and strengthens a result of Pach stating that there is no subgraph‐universal graph in . Furthermore, we characterise for which subdivided stars there is a topological minor‐universal graph in the class of all countable ‐free graphs.