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Perfect codes in quartic Cayley graphs of generalized dihedral groups

Chengcheng Dong, Yuefeng Yang, Changchang Dong

TL;DR

This work classifies connected quartic Cayley graphs on generalized dihedral groups that admit a perfect code and determines all such codes. The authors split the analysis by the size of the intersection |S∩tA|, obtaining nonexistence results for |S∩tA|∈{1,3}, and a detailed existence/structure theory for |S∩tA|=2 or 4. They reduce certain cases to abelian Cayley graphs and leverage known results, while providing explicit constructions of perfect codes (notably when |S∩tA|=4) via layered cycles and a union-structure D with parameters tied to orders in A. The resulting two main theorems give a comprehensive, formula-driven classification and explicit descriptions of all perfect codes, advancing the understanding of efficient domination in Cayley graphs beyond abelian settings. These findings have implications for network coding and combinatorial design in algebraically structured graphs.

Abstract

For a graph $Γ=(VΓ,EΓ)$, a subset $D$ of $VΓ$ is a perfect code in $Γ$ if every vertex of $Γ$ is dominated by exactly one vertex in $D$. In this paper, we classify all connected quartic Cayley graphs on generalized dihedral groups admitting a perfect code, and determine all perfect codes in such graphs.

Perfect codes in quartic Cayley graphs of generalized dihedral groups

TL;DR

This work classifies connected quartic Cayley graphs on generalized dihedral groups that admit a perfect code and determines all such codes. The authors split the analysis by the size of the intersection |S∩tA|, obtaining nonexistence results for |S∩tA|∈{1,3}, and a detailed existence/structure theory for |S∩tA|=2 or 4. They reduce certain cases to abelian Cayley graphs and leverage known results, while providing explicit constructions of perfect codes (notably when |S∩tA|=4) via layered cycles and a union-structure D with parameters tied to orders in A. The resulting two main theorems give a comprehensive, formula-driven classification and explicit descriptions of all perfect codes, advancing the understanding of efficient domination in Cayley graphs beyond abelian settings. These findings have implications for network coding and combinatorial design in algebraically structured graphs.

Abstract

For a graph , a subset of is a perfect code in if every vertex of is dominated by exactly one vertex in . In this paper, we classify all connected quartic Cayley graphs on generalized dihedral groups admitting a perfect code, and determine all perfect codes in such graphs.

Paper Structure

This paper contains 4 sections, 14 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. $|S\cap tA|\neq4$
  3. $|S\cap tA|=4$
  4. Proofs of Theorems \ref{['1.1']} and \ref{['1.2']}

Key Result

Theorem 1.1

Let $\Gamma=\mathrm {Cay}(G,S)$ be a connected quartic Cayley graph on a generalized dihedral group. Without loss of generality assume $t\in S$. Then $\Gamma$ admits a perfect code if and only if one of the following holds: Here, $s_0,s_1,s_2\in A$, $o(s_0)=n$, $m$ is the minimum positive integer such that $s_1^m\in\langle s_0\rangle$ and $l$ is the minimum positive integer such that $s_2^l\in \l

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 18 more