Hypercube minor-universality

TL;DR

The paper establishes sharp-contrast bounds for hypercube minor-universality: Q_d is Ω(2^d/d)-minor-universal but not O(2^d/√d)-minor-universal, refining prior expander-based bounds and tightening the dependence on dimension. Central to the approach is a combinatorial-embedding framework that translates minor containment into structured embeddings, with the box permutation decomposition providing a key tool to realize large families of embeddings inside product graphs. The authors extend the hypercube result to a general product-graph setting, proving both lower and upper bounds for m-minor-universality in complex graph products via a multi-step embedding plan and a careful handling of degree constraints. The work thus connects minor theory, combinatorial embeddings, and product graphs, and opens questions about tightness, transitive graphs, and broader product structures. The constructions employ explicit decompositions, subdivision tricks, and sphere-cut arguments to control embedding complexity and bound minor sizes.

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 $Ω\left(\frac{2^d}{d}\right)$-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 $σ: X \to X$ can be expressed as $σ= σ_1 \circ \dots \circ σ_{2d-1}$, where each $σ_i$ is a one-dimensional permutation, meaning it fixes all coordinates except possibly one. We discuss future directions and pose open problems.

Figures (4)

• Figure 1: There is a combinatorial embedding of the triangle into the star graph, but the triangle is not a minor of the star graph.
• Figure 2: Well-dispersed lemma example.
• Figure 3: Well-dispersed lemma proof strategy.
• Figure 4: $d=2$ strategy.

Theorems & Definitions (36)

• Definition 1.1
• Theorem A: hypercube minor-universality
• Theorem B: product graph minor-universality
• Proposition 1.2: box permutation decomposition
• Definition 2.1: combinatorial embedding
• Lemma 2.2
• proof
• Definition 2.3
• Lemma 2.4
• proof