Total perfect codes in Cayley graphs

Abstract

A total perfect code in a graph $\Gamma$ is a subset $C$ of $V(\Gamma)$ such that every vertex of $\Gamma$ is adjacent to exactly one vertex in $C$. We give necessary and sufficient conditions for a conjugation-closed subset of a group to be a total perfect code in a Cayley graph of the group. As an application we show that a Cayley graph on an elementary abelian $2$-group admits a total perfect code if and only if its degree is a power of $2$. We also obtain necessary conditions for a Cayley graph of a group with connection set closed under conjugation to admit a total perfect code.Comment: This is the final version published in Designs, Codes and Cryptograph

Perfect codes in circulant graphs

“…In [14] it was proved that a conjugation-closed subset C of a group G is a perfect code in a Cayley graph on G if and only if there exists a covering projection from the Cayley graph to a complete graph with C as a fibre. 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.…”

Subgroup perfect codes in Cayley sum graphs

“…For the past few years, perfect codes in Cayley graphs have attracted considerable attention, see, for example, [11,27,28]. In [14], Huang, Xia and Zhou first introduced the concept of a perfect code of a group G. A subset C of G is said to be a perfect code of G if C is a perfect code of some Cayley graph of G. In particular, a subgroup is said to be a subgroup perfect code of G if the subgroup is also a perfect code of G. Also in [14], they gave a necessary and sufficient condition for a normal subgroup of a group G to be a subgroup perfect code of G, and determined all the subgroup perfect codes of dihedral groups and some abelian groups.…”

Perfect Codes in Cayley Sum Graphs

“…In the past a few years, perfect codes in Cayley graphs have attracted considerable attention, see, for example, [11,28,29]. In [14], Huang, Xia and Zhou first introduced the concept of a perfect code of a group G. A subset C of G is said to be a perfect code of G if C is a perfect code of some Cayley graph of G. In particular, a subgroup is said to be a subgroup perfect code of G if the subgroup is also a perfect code of G. Also in [14], they gave a necessary and sufficient condition for a normal subgroup of a group G to be a subgroup perfect code of G, and determined all the subgroup perfect codes of dihedral groups and some abelian groups.…”