Universal graphs for the topological minor relation
Thilo Krill

TL;DR
This paper proves that no single countable planar graph can contain all other countable planar graphs as topological minors, and characterizes when topological minor-universal graphs exist in T-free graph classes.
Contribution
It resolves an open question by showing the non-existence of topological minor-universal graphs in countable planar graphs and characterizes such universality for T-free graphs.
Findings
No topological minor-universal graph exists for all countable planar graphs.
Characterization of subdivided stars T for which T-free graphs have a topological minor-universal graph.
Strengthens previous results on subgraph-universality in planar graphs.
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 K\"uhn 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.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Advanced Graph Theory Research · Limits and Structures in Graph Theory
