Extremal density for sparse minors and subdivisions

Abstract: How dense a graph has to be to necessarily contain (topological) minors of a given graph $H$? When $H$ is a complete graph, this question is well understood by result of Kostochka/Thomason for clique minor, and result of Bollobas-Thomason/Komlos-Szemeredi for topological minor. The situation is a lot less clear when $H$ is a sparse graph. We will present a general result on optimal density condition forcing (topological) minors of a wide range of sparse graphs. As corollaries, we resolve several questions of Reed and Wood on embedding sparse minors. Among others,

- $(1+o(1))t^2$ average degree is sufficient to force the $t\times t$ grid as a topological minor;

- $(3/2+o(1))t$ 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;

- 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.

Joint work with John Haslegrave and Jaehoon Kim.

combinatorics

Audience: researchers in the topic

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Warwick Combinatorics Seminar

Series comments: This is the online combinatorics seminar at Warwick.

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