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Perfect codes and total perfect codes in intersection graphs of finite groups.

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Title: Perfect codes and total perfect codes in intersection graphs of finite groups.
Authors: Wei, Lina1 (AUTHOR), Wang, Xiaomeng1 (AUTHOR), Bian, Hong2 (AUTHOR), Xu, Shou-Jun1 (AUTHOR) shjxu@lzu.edu.cn
Source: Graphs & Combinatorics. Aug2026, Vol. 42 Issue 4, p1-16. 16p.
Subjects: Intersection graph theory, Finite groups, Modular groups, Abelian groups
Abstract: Let Γ be a graph with vertex set V (Γ) . A subset C of V (Γ) is a perfect code of Γ if C is an independent set in Γ such that every vertex in V (Γ) \ C is adjacent to exactly one vertex in C. A subset T of V (Γ) is a total perfect code of Γ if every vertex of Γ is adjacent to exactly one vertex in T. Let G be a group with identity element e. The intersection graph of G, denoted by Γ (G) , is the graph whose vertex set consists of all nontrivial proper subgroups of G, and two distinct vertices H and K are adjacent if and only if H ∩ K ≠ { e } . In this paper, we establish necessary and sufficient conditions for the intersection graphs of finite abelian groups, generalized quaternion groups, and modular groups to have perfect codes and total perfect codes. We characterize dihedral groups and quasi-dihedral groups whose intersection graphs have perfect codes, and prove that the intersection graphs of dihedral groups and quasi-dihedral groups have no total perfect code. Furthermore, we explicitly provide some of the existing perfect codes and total perfect codes in the intersection graphs mentioned above. [ABSTRACT FROM AUTHOR]
Copyright of Graphs & Combinatorics is the property of Springer Nature 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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DbLabel: Engineering Source
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  Data: Perfect codes and total perfect codes in intersection graphs of finite groups.
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  Data: <searchLink fieldCode="AR" term="%22Wei%2C+Lina%22">Wei, Lina</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Wang%2C+Xiaomeng%22">Wang, Xiaomeng</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Bian%2C+Hong%22">Bian, Hong</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Xu%2C+Shou-Jun%22">Xu, Shou-Jun</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> shjxu@lzu.edu.cn</i>
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  Data: <searchLink fieldCode="JN" term="%22Graphs+%26+Combinatorics%22">Graphs & Combinatorics</searchLink>. Aug2026, Vol. 42 Issue 4, p1-16. 16p.
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  Data: <searchLink fieldCode="DE" term="%22Intersection+graph+theory%22">Intersection graph theory</searchLink><br /><searchLink fieldCode="DE" term="%22Finite+groups%22">Finite groups</searchLink><br /><searchLink fieldCode="DE" term="%22Modular+groups%22">Modular groups</searchLink><br /><searchLink fieldCode="DE" term="%22Abelian+groups%22">Abelian groups</searchLink>
– Name: Abstract
  Label: Abstract
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  Data: Let Γ be a graph with vertex set V (Γ) . A subset C of V (Γ) is a perfect code of Γ if C is an independent set in Γ such that every vertex in V (Γ) \ C is adjacent to exactly one vertex in C. A subset T of V (Γ) is a total perfect code of Γ if every vertex of Γ is adjacent to exactly one vertex in T. Let G be a group with identity element e. The intersection graph of G, denoted by Γ (G) , is the graph whose vertex set consists of all nontrivial proper subgroups of G, and two distinct vertices H and K are adjacent if and only if H ∩ K ≠ { e } . In this paper, we establish necessary and sufficient conditions for the intersection graphs of finite abelian groups, generalized quaternion groups, and modular groups to have perfect codes and total perfect codes. We characterize dihedral groups and quasi-dihedral groups whose intersection graphs have perfect codes, and prove that the intersection graphs of dihedral groups and quasi-dihedral groups have no total perfect code. Furthermore, we explicitly provide some of the existing perfect codes and total perfect codes in the intersection graphs mentioned above. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Graphs & Combinatorics is the property of Springer Nature 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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      – Type: doi
        Value: 10.1007/s00373-026-03048-2
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 16
        StartPage: 1
    Subjects:
      – SubjectFull: Intersection graph theory
        Type: general
      – SubjectFull: Finite groups
        Type: general
      – SubjectFull: Modular groups
        Type: general
      – SubjectFull: Abelian groups
        Type: general
    Titles:
      – TitleFull: Perfect codes and total perfect codes in intersection graphs of finite groups.
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            NameFull: Wei, Lina
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            NameFull: Wang, Xiaomeng
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            NameFull: Bian, Hong
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            NameFull: Xu, Shou-Jun
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          Dates:
            – D: 01
              M: 08
              Text: Aug2026
              Type: published
              Y: 2026
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              Value: 42
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            – TitleFull: Graphs & Combinatorics
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