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

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Title: On induced subgraphs of H(n, 3) with maximum degree 1.
Authors: Potechin, Aaron1, Tsang, Hing Yin1
Source: Discrete Mathematics & Theoretical Computer Science (DMTCS). 2026, Vol. 28 Issue 2, p1-41. 41p.
Subjects: Independent sets, Subgraphs, Discrete mathematics, Combinatorics, Graph theory
Abstract: In this paper, we consider induced subgraphs of the Hamming graph H(n, 3). We show that if U ⊆ Z n 3 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| ≤ 3 n−1 + 1. Moreover, all such U with size 3 n−1 + 1 are isomorphic to each other. 2. For n ≥ 6, there exists such a U with size |U| = 3n−1 + 18 and this is optimal for n = 6. 3. If U ∩ {x, x + e1, x + 2e1} ̸= ϕ for all x ∈ Z n 3 then |U| ≤ 3 n−1 + 81. [ABSTRACT FROM AUTHOR]
Copyright of Discrete Mathematics & Theoretical Computer Science (DMTCS) is the property of Discrete Mathematics & Theoretical Computer Science DMTCS and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: On induced subgraphs of H(n, 3) with maximum degree 1.
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  Data: In this paper, we consider induced subgraphs of the Hamming graph H(n, 3). We show that if U ⊆ Z n 3 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| ≤ 3 n−1 + 1. Moreover, all such U with size 3 n−1 + 1 are isomorphic to each other. 2. For n ≥ 6, there exists such a U with size |U| = 3n−1 + 18 and this is optimal for n = 6. 3. If U ∩ {x, x + e1, x + 2e1} ̸= ϕ for all x ∈ Z n 3 then |U| ≤ 3 n−1 + 81. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Discrete Mathematics & Theoretical Computer Science (DMTCS) is the property of Discrete Mathematics & Theoretical Computer Science DMTCS and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – Code: eng
        Text: English
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        PageCount: 41
        StartPage: 1
    Subjects:
      – SubjectFull: Independent sets
        Type: general
      – SubjectFull: Subgraphs
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      – SubjectFull: Discrete mathematics
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      – SubjectFull: Combinatorics
        Type: general
      – SubjectFull: Graph theory
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      – TitleFull: On induced subgraphs of H(n, 3) with maximum degree 1.
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              Text: 2026
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              Y: 2026
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