Isoperimetry in product graphs

TL;DR

This paper establishes a sharp edge-isoperimetric inequality for product graphs, revealing that certain subsets minimize edge-boundary sizes and improving existing inequalities for regular graph products.

Contribution

It introduces a new sharp edge-isoperimetric inequality applicable to all product graphs, extending and strengthening previous results.

Findings

Identifies subsets with minimal edge-boundary in product graphs.

Improves existing inequalities for regular graph products.

Answers open questions on edge-isoperimetry in powers of regular graphs.

Abstract

In this short note, we establish an edge-isoperimetric inequality for arbitrary product graphs. Our inequality is sharp for subsets of many different sizes in every product graph. In particular, it implies that the $2^d$-element sets with smallest edge-boundary in the hypercube are subcubes and is only marginally weaker than the Bollob\'as$\unicode{x2013}$Leader edge-isoperimetric inequalities for grids and tori. Additionally, it improves two edge-isoperimetric inequalities for products of regular graphs proved by Erde, Kang, Krivelevich, and the first author and answers two questions about edge-isoperimetry in powers of regular graphs raised in their work.