On Induced Subgraphs of the Hamming Graph

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).