Universal planar graphs for the topological minor relation

Abstract

Huynh et al. recently showed that a countable graph $G$ which contains every countable planar graph as a subgraph must contain arbitrarily large finite complete graphs as topological minors, and an infinite complete graph as a minor. We strengthen this result by showing that the same conclusion holds, if $G$ contains every countable planar graph as a topological minor. In particular, there is no countable planar graph containing every countable planar graph as a topological minor, answering a question by Diestel and K\"uhn. Moreover, we construct a locally finite planar graph which contains every locally finite planar graph as a topological minor. This shows that in the above result it is not enough to require that $G$ contains every locally finite planar graph as a topological minor.