Total perfect codes in Cayley graphs

TL;DR

This paper characterizes when conjugation-closed subsets form total perfect codes in Cayley graphs, showing that such codes exist in elementary abelian 2-groups only when the degree is a power of two.

Contribution

It provides necessary and sufficient conditions for conjugation-closed subsets to be total perfect codes in Cayley graphs, including a complete characterization for elementary abelian 2-groups.

Findings

Total perfect codes exist in Cayley graphs of elementary abelian 2-groups iff degree is a power of two.

Necessary conditions are established for Cayley graphs with conjugation-closed connection sets to admit total perfect codes.

The paper offers a group-theoretic framework for understanding 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.