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