Perfect codes in Cayley graphs of abelian groups

Peter J. Cameron; Roro Sihui Yap; Sanming Zhou

doi:10.1007/s10623-026-01821-1

Perfect codes in Cayley graphs of abelian groups

Peter J. Cameron
, Roro Sihui Yap
, Sanming Zhou^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

A perfect code in a graph Γ = (V, E) is a subset C of V such that no two vertices in C are adjacent and every vertex in V \C is adjacent to exactly one vertex in C. A total perfect code in Γ is a subset C of V such that every vertex of Γ is adjacent to exactly one vertex in C. In this paper we prove several results on perfect codes and total perfect codes in Cayley graphs of finite abelian groups.

Original language	English
Article number	87
Number of pages	23
Journal	Designs, Codes and Cryptography
Volume	94
Issue number	4
DOIs	https://doi.org/10.1007/s10623-026-01821-1
Publication status	Published - 9 Apr 2026

Keywords

Perfect code
Cayley graph
Tiling of finite groups
Rfficient dominating set
Total perfect code

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Cameron_2026_DCC_Perfect-codes-Cayley-graphs-Abelian-groups_CC
© The Author(s) 2026. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
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author = "Cameron, \{Peter J.\} and Yap, \{Roro Sihui\} and Sanming Zhou",

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AU - Cameron, Peter J.

AU - Yap, Roro Sihui

AU - Zhou, Sanming

N1 - Funding: The third author was supported by a Discovery Project (DP250104965) of the Australian Research Council.

PY - 2026/4/9

Y1 - 2026/4/9

N2 - A perfect code in a graph Γ = (V, E) is a subset C of V such that no two vertices in C are adjacent and every vertex in V \C is adjacent to exactly one vertex in C. A total perfect code in Γ is a subset C of V such that every vertex of Γ is adjacent to exactly one vertex in C. In this paper we prove several results on perfect codes and total perfect codes in Cayley graphs of finite abelian groups.

AB - A perfect code in a graph Γ = (V, E) is a subset C of V such that no two vertices in C are adjacent and every vertex in V \C is adjacent to exactly one vertex in C. A total perfect code in Γ is a subset C of V such that every vertex of Γ is adjacent to exactly one vertex in C. In this paper we prove several results on perfect codes and total perfect codes in Cayley graphs of finite abelian groups.

KW - Perfect code

KW - Cayley graph

KW - Tiling of finite groups

KW - Rfficient dominating set

KW - Total perfect code

U2 - 10.1007/s10623-026-01821-1

DO - 10.1007/s10623-026-01821-1

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JO - Designs, Codes and Cryptography

JF - Designs, Codes and Cryptography

IS - 4

M1 - 87

ER -

Perfect codes in Cayley graphs of abelian groups

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