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In connection with his solution of the Sensitivity Conjecture, Hao Huang(arXiv: 1907.00847, 2019) asked the following question: Given a graph $G$ withhigh symmetry, what can we say about the smallest maximum degree of inducedsubgraphs of $G$ with $\alpha(G)+1$ vertices, where $\alpha(G)$ denotes thesize 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 andmaximum degree at most $\lceil\sqrt{n}\rceil$. Chung et al. proved thisstatement for $k=2$ (the $n$-dimensional cube).

On Induced Subgraphs of the Hamming Graph