Tight Bounds for Hypercube Minor-Universality

TL;DR

This paper addresses the problem of minor-universality for the d-dimensional hypercube $Q_d$, seeking the precise constant-factor threshold in terms of the number of edges. It provides a tighter upper bound by showing a 3-regular expander with $C\cdot\frac{2^d}{d}$ edges cannot be embedded as a minor in $Q_d$, matching the known lower bound up to a constant factor, and thus resolves the tightness question left by prior work. A concise lower-bound proof is offered via a decomposition of permutations of a $d$-dimensional grid into $2d-1$ one-dimensional steps, using a Hall-theorem-based decomposition and a temporal-dimension embedding to realize minors. Together, these results sharpen our understanding of the extremal threshold for minor-universality in hypercubes and have implications for related extremal and structural questions in graph theory.

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.

Figures (1)

• Figure 1: Constructing paths $(x_{i-1},i-1)$→$(x_{i},i)$ and $(z_{i-1},i-1)$→$(z_{i},i)$ when $z_i=x_{i-1}$.

Theorems & Definitions (7)

• Theorem 1.1: Theorem A in BKT25
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Theorem 3.1
• proof