# Complete graph decompositions and p-groupoids

## TL;DR

This paper explores the structure of P-groupoids derived from complete graph decompositions, characterizing their properties and specific examples, including connections to quasigroups and dihedral groups.

## Contribution

It provides a characterization of P-groupoids related to Hamiltonian decompositions and constructs examples from cyclic groups, revealing their algebraic properties.

## Key findings

- Left distributive P-groupoids are distributive quasigroups
- Characterization of P-groupoids for Hamiltonian decompositions of odd prime order graphs
- P-quasigroups from cyclic groups have dihedral group as right multiplication group

## Abstract

We study P-groupoids that arise from certain decompositions of complete graphs. We show that left distributive P-groupoids are distributive, quasigroups. We characterize P-groupoids when the corresponding decomposition is a Hamiltonian decomposition for complete graphs of odd, prime order. We also study a specific example of a P-quasigroup constructed from cyclic groups of odd order. We show such P-quasigroups have characteristic left and right multiplication groups, as well as the right multiplication group is isomorphic to the dihedral group.