Perfect codes in quartic Cayley graphs of generalized dihedral groups

TL;DR

This paper classifies all connected quartic Cayley graphs on generalized dihedral groups that admit perfect codes and determines all such codes, advancing understanding of perfect codes in algebraic graph structures.

Contribution

It provides a complete classification of perfect codes in a specific class of Cayley graphs, which was previously unexplored.

Findings

All connected quartic Cayley graphs on generalized dihedral groups admitting perfect codes are classified.

Explicit descriptions of all perfect codes in these graphs are provided.

The structural conditions for the existence of perfect codes in these graphs are identified.

Abstract

For a graph $\Gamma=(V\Gamma,E\Gamma)$, a subset $D$ of $V\Gamma$ is a perfect code in $\Gamma$ if every vertex of $\Gamma$ 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.