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).

中文翻译：

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关于汉明图的诱导子图

结合他对敏感性猜想的解决方案，Hao Huang (arXiv: 1907.00847, 2019) 提出了以下问题：给定一个具有高度对称性的图 $G$，我们可以说 $G$ 的最小诱导子图的最大度数$\alpha(G)+1$个顶点，其中$\alpha(G)$表示$G$中最大的独立集的大小？我们为 $H(n,k)$ 研究这个问题，$n$ 维汉明图在大小为 $k$ 的字母表上。推广 Chung 等人的结构。(JCT-A, 1988)，我们证明 $H(n,k)$ 有一个诱导子图，其顶点数超过 $\alpha(H(n,k))$，最大度数至多 $\lceil\sqrt{ n}\rceil$。钟等人。证明了 $k=2$（$n$ 维立方体）的这个陈述。