Perfect codes in quartic Cayley graphs of generalized dihedral groups

Abstract

For a graph $\hat{I}“=(V\hat{I}“,E\hat{I}“)$, a subset $D$ of $V\hat{I}“$ is a perfect code in $\hat{I}“$ if every vertex of $\hat{I}“$ 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.