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PreprintCC BY 4.0arXiv

Existence of regular sets in circulant quartic graphs: a complete characterization

  • A. Abdollahi
  • J. Bagherian
  • F. Jafari
  • M. Khatami
  • Z. Shokoohi
  • R. Sobhani
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Abstract

Let Γ=(V,E) be a graph and let a,b be nonnegative integers. A nonempty proper subset C⊂ V is an \emph{(a,b)-regular set} if each vertex of C has exactly a neighbors in C and each vertex outside C has exactly b neighbors in C. Perfect codes and total perfect codes are the (0,1)- and (1,1)-regular sets. For every connected circulant quartic graph Cay(ℤn,{x± k,x±ℓ}) and every pair (a,b) we decide, in terms of n, k and ℓ, whether an (a,b)-regular set exists. Existence is proved by exhibiting unions of residue classes; non-existence comes from a Fourier-analytic obstruction combined with the fact that certain sums of roots of unity cannot vanish.

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Cite

BibTeXRIS
BibTeX
@misc{abdollahi2026existence,
  title = {{Existence of regular sets in circulant quartic graphs: a complete characterization}},
  author = {A. Abdollahi and J. Bagherian and F. Jafari and M. Khatami and Z. Shokoohi and R. Sobhani},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.09414},
  eprint = {2609.09414},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09414},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Existence of regular sets in circulant quartic graphs: a complete characterization
AU  - A. Abdollahi
AU  - J. Bagherian
AU  - F. Jafari
AU  - M. Khatami
AU  - Z. Shokoohi
AU  - R. Sobhani
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.09414
UR  - https://arxiv.org/abs/2609.09414
AB  - Let Γ=(V,E) be a graph and let a,b be nonnegative integers. A nonempty proper subset C⊂ V is an \emph{(a,b)-regular set} if each vertex of C has exactly a neighbors in C and each vertex outside C has exactly b neighbors in C. Perfect codes and total perfect codes are the (0,1)- and (1,1)-regular sets. For every connected circulant quartic graph Cay(ℤ_n,{x^± k,x^±ℓ}) and every pair (a,b) we decide, in terms of n, k and ℓ, whether an (a,b)-regular set exists. Existence is proved by exhibiting unions of residue classes; non-existence comes from a Fourier-analytic obstruction combined with the fact that certain sums of roots of unity cannot vanish.
ER  -