Sensitivity and Hamming graphs

TL;DR

This work extends Huang’s sensitivity framework from the Boolean hypercube to the $m$-ary Hamming graphs $H(n,m)$, introducing the partition-parameters $\Delta(\Pi)$ and $\iota(\Pi)$ to study low-degree partitions. It provides constructive imbalanced partitions with $\Delta(\Pi)\le d$ and large imbalance, disproving the Strong $m$-ary Sensitivity Conjecture while establishing a weaker bound $s(f) \ge \sqrt{\deg(f)/(m-1)}$ for $m$-ary functions, thereby connecting sensitivity to degree in a broad setting. The results leverage lifting methods, supersaturation, and connections to covering codes and abelian Cayley graphs to bound subgraph degrees and to relate subgraph structure to sensitivity. Together, these contributions advance understanding of how symmetry and combinatorial structure in $H(n,m)$ constrain sensitivity and degree, with implications for generalized sensitivity conjectures and complexity measures beyond the Boolean case.

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, García-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.

Figures (1)

• Figure 1: The construction in Proposition \ref{['prop:FirstConstruction']} with $m=4$ and $n=3$

Theorems & Definitions (18)

• Theorem 1.1: Imbalanced partitions
• Theorem 1.2: Sensitivity
• Proposition 2.1: Imbalanced partitions of degree $1$
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.3: Imbalanced partitions
• proof