# Isoperimetry, Stability, and Irredundance in Direct Products

## TL;DR

This paper establishes optimal isoperimetric inequalities and stability results for independent sets in direct products of complete multipartite graphs, leading to a proof that the upper irredundance number equals the independence number in most cases, confirming a significant conjecture.

## Contribution

It introduces the first optimal vertex isoperimetric inequality for direct products of complete multipartite graphs and applies it to prove a near-complete validation of Burcroff's conjecture.

## Key findings

- Optimal isoperimetric inequality for direct products of complete multipartite graphs.
- Stability result showing large independent sets are close to maximal.
- Upper irredundance number equals independence number in all but 37 cases.

## Abstract

The direct product of graphs $G_1,\ldots,G_n$ is the graph with vertex set $V(G_1)\times\cdots\times V(G_n)$ in which two vertices $(g_1,\ldots,g_n)$ and $(g_1',\ldots,g_n')$ are adjacent if and only if $g_i$ is adjacent to $g_i'$ in $G_i$ for all $i$. Building off of the recent work of Brakensiek, we prove an optimal vertex isoperimetric inequality for direct products of complete multipartite graphs. Applying this inequality, we derive a stability result for independent sets in direct products of balanced complete multipartite graphs, showing that every large independent set must be close to the maximal independent set determined by setting one of the coordinates to be constant. Armed with these isoperimetry and stability results, we prove that the upper irredundance number of a direct product of balanced complete multipartite graphs is equal to its independence number in all but at most $37$ cases. This proves most of a conjecture of Burcroff that arose as a strengthening of a conjecture of the second author and Iyer. We also propose a further strengthening of Burcroff's conjecture.