Total perfect codes in Cayley graphs

Abstract

A total perfect code in a graph $Γ$ is a subset $C$ of $V(Γ)$ such that every vertex of $Γ$ 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.

This is the final version published in: Designs, Codes and Cryptography 81 (2016) 489-504. The surname of the first author of [8] in the previous version (which is [7] in this version) was mistakenly spelt as Dejtera. The correct form should be Dejter, and in the present version this typo has been corrected