On induced subgraphs of $H(n,3)$ with maximum degree $1$

TL;DR

This paper investigates the structure and size limitations of induced subgraphs with maximum degree 1 in the Hamming graph $H(n,3)$, establishing bounds and characterizing extremal configurations.

Contribution

It provides new bounds on the size of such subgraphs and characterizes their structure, especially when disjoint from maximum independent sets.

Findings

Maximum size of such subgraphs is $3^{n-1}+1$ when disjoint from maximum independent sets.

Existence of larger subgraphs with size $3^{n-1}+18$ for $n \\geq 6$.

Upper bound of $3^{n-1}+81$ for subgraphs intersecting every triple of the form $\{x, x+e_1, x+2e_1\}$.

Abstract

In this paper, we consider induced subgraphs of the Hamming graph $H(n,3)$. We show that if $U \subseteq \mathbb{Z}_3^n$ and $U$ induces a subgraph of $H(n,3)$ with maximum degree at most $1$ then 1. If $U$ is disjoint from a maximum size independent set of $H(n,3)$ then $|U| \leq 3^{n-1}+1$. Moreover, all such $U$ with size $3^{n-1}+1$ are isomorphic to each other. 2. For $n \geq 6$, there exists such a $U$ with size $|U| = 3^{n-1}+18$ and this is optimal for $n = 6$. 3. If $U \cap \{x, x+e_1, x+2e_1\} \ne \phi$ for all $x \in \mathbb{Z}_3^n$ then $|U| \leq 3^{n-1} + 81$.