Universality in minor-closed graph classes [article]

Tony Huynh and Bojan Mohar and Robert Šámal and Carsten Thomassen and David R. Wood
2021 arXiv   pre-print
Stanislaw 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,
more » ... 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).
arXiv:2109.00327v1 fatcat:wmv5mwtfvjfttfhjmxuswpn27e