On Induced Subgraphs of the Hamming Graph

TL;DR

This paper investigates the structure of induced subgraphs in high-symmetry graphs, specifically Hamming graphs, establishing bounds on their maximum degree relative to their size, extending previous results from hypercubes to more general cases.

Contribution

It generalizes a known construction for hypercubes to Hamming graphs with larger alphabets, providing bounds on the maximum degree of induced subgraphs with size just above the independence number.

Findings

H(n,k) has an induced subgraph with more than α(H(n,k)) vertices and maximum degree at most ⌈√n⌉.

Extension of Chung et al.'s result from k=2 to larger alphabets.

Provides insights into the structure of symmetric graphs related to the Sensitivity Conjecture.

Abstract

In connection with his solution of the Sensitivity Conjecture, Hao Huang (arXiv: 1907.00847, 2019) asked the following question: Given a graph $G$ with high symmetry, what can we say about the smallest maximum degree of induced subgraphs of $G$ with $\alpha(G)+1$ vertices, where $\alpha(G)$ denotes the size of the largest independent set in $G$? We study this question for $H(n,k)$, the $n$-dimensional Hamming graph over an alphabet of size $k$. Generalizing a construction by Chung et al. (JCT-A, 1988), we prove that $H(n,k)$ has an induced subgraph with more than $\alpha(H(n,k))$ vertices and maximum degree at most $\lceil\sqrt{n}\rceil$. Chung et al. proved this statement for $k=2$ (the $n$-dimensional cube).