Sensitivity and Hamming graphs

TL;DR

This paper investigates the structure of Hamming graphs, showing they can be partitioned into low-degree subgraphs, and explores sensitivity conjectures, proving a weaker form and disproving a stronger one.

Contribution

It provides a new partitioning result for Hamming graphs and resolves the Strong $m$-ary Sensitivity Conjecture by disproving it, while proving a weaker version.

Findings

Hamming graphs admit imbalanced partitions into low-degree subgraphs for all m ≥ 3

Disproves the Strong m-ary Sensitivity Conjecture

Proves the weaker m-ary Sensitivity Conjecture with polynomial bounds

Abstract

For any $m\geq 3$ we show that the Hamming graph $H(n,m)$ admits an imbalanced partition into $m$ sets, each inducing a subgraph of low maximum degree. This improves previous results by Tandya and by Potechin and Tsang, and disproves the Strong $m$-ary Sensitivity Conjecture of Asensio, Garc\'ia-Marco, and Knauer. On the other hand, we prove their weaker $m$-ary Sensitivity Conjecture by showing that the sensitivity of any $m$-ary function is bounded from below by a polynomial expression in its degree.