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 Kühn. 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.
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Much of the research leading to the results presented in this paper was carried out while the author was supported by the Austrian Science Fund (FWF) Grant no. P31889-N35.
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Lehner, F. Universal Planar Graphs for the Topological Minor Relation. Combinatorica 44, 209–230 (2024). https://doi.org/10.1007/s00493-023-00073-0
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DOI: https://doi.org/10.1007/s00493-023-00073-0