Hypercube minor-universality

TL;DR

This paper investigates the minor-universality of hypercubes, establishing bounds on how many edges are needed for hypercubes to contain all smaller graphs as minors, and introduces a decomposition method for permutations of multi-dimensional boxes.

Contribution

It proves bounds on the minor-universality of hypercubes and introduces a novel permutation decomposition technique for multi-dimensional grids.

Findings

Hypercube $Q_d$ is $rac{2^d}{d}$-minor-universal.

Hypercube $Q_d$ is not $rac{C2^d}{ oot d}$-minor-universal for some constant C.

Permutation of a multi-dimensional box can be decomposed into $2d-1$ one-dimensional permutations.

Abstract

A graph $G$ is $m$-minor-universal if every graph with at most $m$ edges (and no isolated vertices) is a minor of $G$. We prove that the $d$-dimensional hypercube, $Q_d$, is $\Omega\left(\frac{2^d}{d}\right)$-minor-universal, and that there exists an absolute constant $C >0$ such that $Q_d$ is not $\frac{C2^d}{\sqrt{d}}$-minor-universal. Similar results are obtained in a more generalized setting, where we bound the size of minors in a product of finite connected graphs. A key component of our proof is the following claim regarding the decomposition of a permutation of a box into simpler, one-dimensional permutations: Let $n_1, \dots, n_d$ be positive integers, and define $X := [n_1] \times \dots \times [n_d]$. We prove that every permutation $\sigma: X \to X$ can be expressed as $\sigma = \sigma_1 \circ \dots \circ \sigma_{2d-1}$, where each $\sigma_i$ is a one-dimensional permutation, meaning it fixes all coordinates except possibly one. We discuss future directions and pose open problems.