DeepAI

# Large Minors in Expanders

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.

12/21/2022

### Edge separators for graphs excluding a minor

We prove that every n-vertex K_t-minor-free graph G of maximum degree Δ ...
02/20/2018

### Wireless Expanders

This paper introduces an extended notion of expansion suitable for radio...
12/06/2018

### Euler Transformation of Polyhedral Complexes

We propose an Euler transformation that transforms a given d-dimensional...
12/25/2021

### Modularity and edge sampling

Suppose that there is an unknown underlying graph G on a large vertex se...
02/19/2018

### Discrepancy Analysis of a New Randomized Diffusion Algorithm for Weighted Round Matrices

For an arbitrary initial configuration of indivisible loads over vertice...
12/06/2020

### The Local Structure of Bounded Degree Graphs

Let G=(V,E) be a simple graph with maximum degree d. For an integer k∈ℕ,...
12/28/2018

### Occupancy fraction, fractional colouring, and triangle fraction

Given ε>0, there exists f_0 such that, if f_0 < f <Δ^2+1, then for any g...