Universal planar graphs for the topological minor relation

TL;DR

This paper investigates the properties of universal planar graphs, showing that containing all countable planar graphs as topological minors implies the presence of large complete minors, and constructs a locally finite planar graph that is topologically universal for locally finite planar graphs.

Contribution

It strengthens previous results by proving that universal planar graphs must contain large complete minors and constructs a locally finite planar graph that is topologically universal for all locally finite planar graphs.

Findings

Universal planar graphs contain arbitrarily large complete minors.

No countable planar graph contains all others as topological minors.

Constructed a locally finite planar graph that contains all locally finite planar graphs as topological minors.

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.