An induced subgraph of the Hamming graph with maximum degree 1

For every graph $G$, let $\alpha(G)$ denote its independence number. What is the minimum of the maximum degree of an induced subgraph of $G$ with $\alpha(G)+1$ vertices? We study this question for the $n$-dimensional Hamming graph over an alphabet of size $k$. In this paper, we give a construction to prove that the answer is $1$ for all $n$ and $k$ with $k \geq 3$. This is an improvement over an earlier work showing that the answer is at most $\lceil \sqrt{n} \, \rceil$.