Generalized Cayley graphs and perfect code

Abstract

A subset C of the vertex set of a graph Γ is said to be $(a,b)$-regular if C induces an a-regular subgraph and every vertex outside C is adjacent to exactly b vertices in C. A $(0,1)$-regular set is called a perfect code. Let G be a group and $\alpha \in \mathrm{Aut}\left(G\right)$ such that ${\alpha }^2=\mathrm{id}$. Let $S\subseteq G$, with $\alpha \left(S\right)=S^{-1}$ and $S\cap \left\{\alpha \right(g^{-1})g:g\in G\}=\varnothing$. The generalized Cayley graph of $(G,\alpha )$ with respect to S is a graph with vertex set G and two distinct elements $g,h\in G$ are adjacent if and only if $\alpha \left(g^{-1}\right)h\in S$. If $\alpha =\mathrm{id}$, then the described graph is called a Cayley graph of G with respect to S. By an $(a,b)$-regular set (resp. a perfect code) of $(G,\alpha )$ we mean an $(a,b)$-regular set (resp. a perfect code) in a generalized Cayley graph of $(G,\alpha )$ with respect to some subset S. Let G be a group, $\alpha \in \mathrm{Aut}\left(G\right)$, ${\alpha }^2=\mathrm{id}$ and H be a subgroup of G. In this paper, we give a necessary and sufficient condition for H to be a perfect code of $(G,\alpha )$. This result is a generalization of [9, Theorem 3.1] that gives a condition for a subgroup to be a perfect code in a Cayley graph of G. As another result, when G is an abelian group, we determine all pairs $(a,b)$ such that H is an $(a,b)$-regular set of $(G,\alpha )$.