Tight Bounds for Hypercube Minor-Universality

TL;DR

This paper establishes tight bounds on the minor-universality of hypercubes, showing they do not contain certain expanders as minors, thus refining understanding of hypercube minor-embedding limits.

Contribution

It proves that hypercubes do not contain 3-uniform expanders with a certain number of edges as minors, matching previous lower bounds and answering an open question.

Findings

Hypercubes do not contain certain expanders as minors.

Results match the known lower bounds up to a constant factor.

Answers an open question about hypercube minor-universality.

Abstract

Benjamini, Kalifa and Tzalik recently proved that there is an absolute constant $c>0$ such that any graph with at most $c\cdot2^d/d$ edges and no isolated vertices is a minor of the $d$-dimensional hypercube $Q_d$, while there is an absolute constant $K > 0$ such that $Q_d$ is not $(K\cdot2^d/\sqrt{d})$-minor-universal. We show that $Q_d$ does not contain 3-uniform expander graphs with $C\cdot2^d/d$ edges as minors. This matches the lower bound up to a constant factor and answers one of their questions.