Large Minors in Expanders
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In this paper we study expander graphs and their minors. Specifically, we attempt to answer the following question: what is the largest function f(n,α,d), such that every n-vertex α-expander with maximum vertex degree at most d contains every graph H with at most f(n,α,d) edges and vertices as a minor? Our main result is that there is some universal constant c, such that f(n,α,d)≥n/c n·(α/d )^c. This bound achieves a tight dependence on n: it is well known that there are bounded-degree n-vertex expanders, that do not contain any grid with Ω(n/ n) vertices and edges as a minor. The best previous result showed that f(n,α,d) ≥Ω(n/^κn), where κ depends on both α and d. Additionally, we provide a randomized algorithm, that, given an n-vertex α-expander with maximum vertex degree at most d, and another graph H containing at most n/c n·(α/d )^c vertices and edges, with high probability finds a model of H in G, in time poly(n)· (d/α)^O( (d/α) ). We note that similar but stronger results were independently obtained by Krivelevich and Nenadov: they show that f(n,α,d)=Ω(nα^2/d^2 n), and provide an efficient algorithm, that, given an n-vertex α-expander of maximum vertex degree at most d, and a graph H with O( nα^2/d^2 n) vertices and edges, finds a model of H in G. Finally, we observe that expanders are the `most minor-rich' family of graphs in the following sense: for every n-vertex and m-edge graph G, there exists a graph H with O ( n+m/ n) vertices and edges, such that H is not a minor of G.
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