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arXiv · 2203.03477

Universal planar graphs for the topological minor relation

Florian Lehner
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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.

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Florian Lehner. 2022-03-07. Universal planar graphs for the topological minor relation. https://arxiv.org/abs/2203.03477

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