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arXiv · 2012.02159

Extremal density for sparse minors and subdivisions

John Haslegrave·Jaehoon Kim·Hong Liu
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Abstract

We prove an asymptotically tight bound on the extremal density guaranteeing subdivisions of bounded-degree bipartite graphs with a mild separability condition. As corollaries, we answer several questions of Reed and Wood on embedding sparse minors. Among others, $\bullet$ $(1+o(1))t^2$ average degree is sufficient to force the $t\times t$ grid as a topological minor; $\bullet$ $(3/2+o(1))t$ average degree forces every $t$-vertex planar graph as a minor, and the constant $3/2$ is optimal, furthermore, surprisingly, the value is the same for $t$-vertex graphs embeddable on any fixed surface; $\bullet$ a universal bound of $(2+o(1))t$ on average degree forcing every $t$-vertex graph in any nontrivial minor-closed family as a minor, and the constant 2 is best possible by considering graphs with given treewidth.

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John Haslegrave, Jaehoon Kim, Hong Liu. 2020-12-03. Extremal density for sparse minors and subdivisions. https://doi.org/10.1093/imrn%2Frnab154

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