Total perfect codes in Cayley sum graphs of cyclic groups

TL;DR

This paper investigates the existence of total perfect codes in Cayley sum graphs over cyclic groups, establishing algebraic conditions and extending results to direct products of cyclic groups.

Contribution

It introduces necessary and sufficient conditions for total perfect codes in Cayley sum graphs of cyclic groups and generalizes these to products of cyclic groups.

Findings

Conditions for total perfect codes in Cayley sum graphs are characterized.

A correspondence between total perfect codes and group factorizations is established.

Results are extended to direct products of cyclic groups.

Abstract

We consider Cayley sum graphs over the cyclic group $\mathbb{Z}_n$ and aim to explore several necessary and sufficient conditions for the existence of total perfect codes in these graphs. Specifically, we examine various cases for the connection set of the graph including when it is periodic, aperiodic, or square-free. To this end, we utilize a correspondence that we first establish between total perfect codes and factorizations of groups, along with their algebraic properties. We then generalize some of these conditions to the direct product of cyclic groups, i.e. $\mathbb{Z}_{n_1} \times \dots \times \mathbb{Z}_{n_d}$.