Year: 2025
Type: preprint
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. $\mathbb{Z}_{n_1} \times \dots \times \mathbb{Z}_{n_d}$.
Source: arXiv (Cornell University)
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4416619984
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