Perfect Codes in Cayley Graphs

Abstract

Given a graph Γ, a subset C of V (Γ) is called a perfect code in Γ if every vertex of Γ is at distance no more than one to exactly one vertex in C, and a subset C of V (Γ) is called a total perfect code in Γ if every vertex of Γ is adjacent to exactly one vertex in C. In this paper we study perfect codes and total perfect codes in Cayley graphs, with a focus on the following themes: when a subgroup of a given group is a (total) perfect code in a Cayley graph of the group; and how to construct new (total) perfect codes in a Cayley graph from known ones using automorphisms of the underlying group. We prove several results around these questions.

Characterization of subgroup perfect codes in Cayley graphs

“…The problem whether a subgroup is a perfect code of the group has attracted notable attention. Given a normal subgroup H of a group G, a sufficient and necessary condition for H to be a perfect code of G was given in [6] as follows (we have replaced g and gh in the statement of [6, Theorem 2.2] with x and y respectively). As an application of this result, the subgroup perfect codes of cyclic groups were completely determined in [6].…”

“…Given a normal subgroup H of a group G, a sufficient and necessary condition for H to be a perfect code of G was given in [6] as follows (we have replaced g and gh in the statement of [6, Theorem 2.2] with x and y respectively). As an application of this result, the subgroup perfect codes of cyclic groups were completely determined in [6]. It turns out that a subgroup H of a cyclic group G is a perfect code of G if and only if either |H| or |G/H| is odd (see [6,Corollary 2.8]).…”

“…As an application of this result, the subgroup perfect codes of cyclic groups were completely determined in [6]. It turns out that a subgroup H of a cyclic group G is a perfect code of G if and only if either |H| or |G/H| is odd (see [6,Corollary 2.8]). The authors in [6] also classified the subgroup perfect codes of dihedral groups (see [6,Theorem 2.11]).…”

“…It turns out that a subgroup H of a cyclic group G is a perfect code of G if and only if either |H| or |G/H| is odd (see [6,Corollary 2.8]). The authors in [6] also classified the subgroup perfect codes of dihedral groups (see [6,Theorem 2.11]). Recently, the subgroup perfect codes of generalized quaternion groups were classified in [9] (see [9,Theorem 1.7]).…”

Perfect codes in circulant graphs

“…A similar result was obtained in [21] for total perfect codes in Cayley graphs. In a recent work [10], perfect codes in Cayley graphs were studied from the viewpoint of group rings, and among other results conditions for a normal subgroup of a finite group to be a perfect code in some Cayley graph of the group were obtained.…”