Perfect Codes in Cayley Graphs of Hajós Groups

Abstract

A perfect code in a graph Γ is an independent set C such that every vertex of Γ outside C has exactly one neighbour in C. A perfect code in a directed graph can be defined similarly by requiring no arc between vertices of C and for every vertex v outside C there exists exactly one vertex u in C such that the arc from u to v exists in Γ. A subset X of an abelian group G is said to be periodic if there exists a non-identity element g of G such that g + X = X. A factorization of G is a pair of nonempty subsets $(A, B)$ of G such that every element g of G can be expressed uniquely as g = a+b with a ∈ A and b ∈ B. If for every factorization $(A, B)$ of an abelian group G at least one of A and B is periodic, then G is said to be a Hajós group. In this paper we classify all Cayley graphs (directed or undirected) of Hajós groups which admit perfect codes, and moreover we determine all perfect codes in such Cayley graphs.

Subgroup Perfect Codes in Bi-Cayley Graphs
Masoumeh Koohestani, Nasrin Soltankhah, Mohsen Ghasemi
arXiv12 Oct 2026PreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
Perfect state transfer on Cayley graphs over dihedral groups: A complete and practical characterization
Shixin Wang
arXiv31 Aug 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
On subgroup perfect codes and regular sets in generalized Cayley graphs
Yasamin Khaefi, Fateme Sadat Seiedali
arXiv1 Nov 2026PreprintarXiv licenceFull textAlgebra & number theory Combinatorics & discrete mathematicsarXiv
A complete classification of perfect state transfer in Grover walks on abelian Cayley graphs
Koushik Bhakta, Bikash Bhattacharjya
arXiv15 Oct 2026PreprintarXiv licenceFull textCombinatorics & discrete mathematics Quantum physics & quantum informationarXiv
Cayley graphs of abelian groups, circular colorings and discrete fundamental groups
Kengo Enami, Takahiro Matsushita
arXiv26 Oct 2026PreprintarXiv licenceFull textAlgebra & number theory Combinatorics & discrete mathematicsarXiv

Cite

BibTeX RIS

BibTeX

@article{hafidh2026perfect,
  title = {{Perfect Codes in Cayley Graphs of Hajós Groups}},
  author = {Yusuf Hafidh and Binzhou Xia and Sanming Zhou},
  journal = {The Electronic Journal of Combinatorics},
  year = {2026},
  month = oct,
  doi = {10.37236/14502},
  url = {https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p17},
  note = {CC BY 4.0},
}

RIS

TY  - JOUR
TI  - Perfect Codes in Cayley Graphs of Hajós Groups
AU  - Yusuf Hafidh
AU  - Binzhou Xia
AU  - Sanming Zhou
PY  - 2026
DA  - 2026/10/23
JO  - The Electronic Journal of Combinatorics
DO  - 10.37236/14502
UR  - https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p17
AB  - A perfect code in a graph Γ is an independent set C such that every vertex of Γ outside C has exactly one neighbour in C. A perfect code in a directed graph can be defined similarly by requiring no arc between vertices of C and for every vertex v outside C there exists exactly one vertex u in C such that the arc from u to v exists in Γ. A subset X of an abelian group G is said to be periodic if there exists a non-identity element g of G such that g + X = X. A factorization of G is a pair of nonempty subsets $(A, B)$ of G such that every element g of G can be expressed uniquely as g = a+b with a ∈ A and b ∈ B. If for every factorization $(A, B)$ of an abelian group G at least one of A and B is periodic, then G is said to be a Hajós group. In this paper we classify all Cayley graphs (directed or undirected) of Hajós groups which admit perfect codes, and moreover we determine all perfect codes in such Cayley graphs.
SN  - 1077-8926
ER  -

Abstract

Related

Cite