An induced subgraph of the Hamming graph with maximum degree 1

TL;DR

This paper investigates the minimum maximum degree of an induced subgraph with size one more than the independence number in Hamming graphs, providing a construction that achieves degree 1 for all dimensions and alphabet sizes at least 3.

Contribution

It introduces a construction demonstrating that the minimum maximum degree is 1 for all n and k ≥ 3, improving previous bounds.

Findings

Minimum maximum degree is 1 for all n and k ≥ 3

Construction method for such subgraphs

Improves previous upper bound from √n to 1

Abstract

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 $\lceil \sqrt{n} \, \rceil$.