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Universality in minor-closed graph classes

Profile image of Bojan MoharBojan Mohar

2021, Cornell University - arXiv

https://doi.org/10.48550/ARXIV.2109.00327
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72 pages

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Abstract

Stanisław Ulam asked whether there exists a universal countable planar graph (that is, a countable planar graph that contains every countable planar graph as a subgraph). János Pach (1981) answered this question in the negative. We strengthen this result by showing that every countable graph that contains all countable planar graphs must contain (i) an infinite complete graph as a minor, and (ii) a subdivision of the complete graph K t with multiplicity t, for every finite t. On the other hand, we construct a countable graph that contains all countable planar graphs and has several key properties such as linear colouring numbers, linear expansion, and every finite n-vertex subgraph has a balanced separator of size O(√ n). The graph is T 6 P ∞ , where T k is the universal treewidth-k countable graph (which we define explicitly), P ∞ is the 1-way infinite path, and denotes the strong product. More generally, for every positive integer t we construct a countable graph that contains every countable K t-minor-free graph and has the above key properties. Our final contribution is a construction of a countable graph that contains every countable K t-minor-free graph as an induced subgraph, has linear colouring numbers and linear expansion, and contains no subdivision of the countably infinite complete graph (implying (ii) above is best possible).

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