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

Aaron Potechin, Hing Yin Tsang

TL;DR

This work investigates the largest possible subsets U of Z_3^n whose induced subgraph in the Hamming graph H(n,3) has maximum degree 1. The authors develop a dimension-collapsing framework and canonical-structure approach, proving that if U is disjoint from a maximum independent set, then |U| ≤ 3^{n-1}+1 with a unique extremal configuration up to automorphism, while removing this restriction allows larger constructions (up to 3^{n-1}+18 for n≥6). A key contribution is an upper bound for i-saturated 1-degree subgraphs, showing |U| ≤ 3^{n-1}+81, derived via a SAT-verified line-extension lemma and a canonical-growth argument that yields a large canonical substructure of dimension at least n−4. Together, these results illuminate the tightness and structure of extremal induced subgraphs in H(n,3) and connect to broader questions about Huang-type bounds in Hamming graphs. The methods combine combinatorial structure, SAT-solving verification, and dimension-reduction techniques, with implications for understanding near-half-size induced subgraphs in product graphs.

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 $U \cap \{x, x+e_1, x+2e_1\} \ne φ$ for all $x \in \mathbb{Z}_3^n$ then $|U| \leq 3^{n-1} + 81$.

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

TL;DR

This work investigates the largest possible subsets U of Z_3^n whose induced subgraph in the Hamming graph H(n,3) has maximum degree 1. The authors develop a dimension-collapsing framework and canonical-structure approach, proving that if U is disjoint from a maximum independent set, then |U| ≤ 3^{n-1}+1 with a unique extremal configuration up to automorphism, while removing this restriction allows larger constructions (up to 3^{n-1}+18 for n≥6). A key contribution is an upper bound for i-saturated 1-degree subgraphs, showing |U| ≤ 3^{n-1}+81, derived via a SAT-verified line-extension lemma and a canonical-growth argument that yields a large canonical substructure of dimension at least n−4. Together, these results illuminate the tightness and structure of extremal induced subgraphs in H(n,3) and connect to broader questions about Huang-type bounds in Hamming graphs. The methods combine combinatorial structure, SAT-solving verification, and dimension-reduction techniques, with implications for understanding near-half-size induced subgraphs in product graphs.

Abstract

In this paper, we consider induced subgraphs of the Hamming graph . We show that if and induces a subgraph of with maximum degree at most then 1. If is disjoint from a maximum size independent set of then . Moreover, all such with size are isomorphic to each other. 2. For , there exists such a with size and this is optimal for . 3. If for all then .
Paper Structure (24 sections, 34 theorems, 18 equations, 3 figures)

This paper contains 24 sections, 34 theorems, 18 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Our Results
  3. Organization
  4. Preliminaries
  5. Collapsing one dimension of ${\mathbb Z}_3^n$
  6. Canonical Sets
  7. Uniqueness of canonical sets
  8. Canonical Paths
  9. Size lower bound
  10. Finding subsets via SAT solvers
  11. Examples of subsets with 6 and 18 extra points
  12. Size upper bound
  13. Proof strategy for our size upper bound
  14. The line extension lemma
  15. $n = 2$
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $U \subseteq {\mathbb Z}_3^n$. If $U$ is disjoint from a maximum size independent set of $H(n,3)$ and the induced subgraph of $H(n,3)$ on $U$ has maximum degree at most $1$ then $|U| \le 3^{n-1}+1$.

Figures (3)

  • Figure 1: This figure shows the independent set $A_3$. Note that the first block of $A_3$ is $A_2$, the second block of $A_3$ is $C_2$, and the third block of $A_3$ is $B_2$.
  • Figure 2: This figure shows $A$, $B$, $C$, $X$, $Y$, and $Z$.
  • Figure 3: This figure shows $D_2$ and $D_3$.

Theorems & Definitions (85)

  • Remark
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: Hamming graph $H(n,3)$
  • Definition 2.2: Standard basis for ${\mathbb Z}_3^n$
  • Definition 2.3: All zero point
  • Definition 2.4: Induced degree
  • Example 1
  • Proposition 2.5
  • ...and 75 more