Parastrophically uncancellable quasigroup equations

Abstract

Krstić initiated the use of cubic graphs in solving quasigroup equations. Based on his work, Krapež and Živković proved that there is a bijective correspondence between classes of parastrophically equivalent parastrophically uncancellable generalized quadratic functional equations on quasigroups and three-connected cubic (multi)graphs. We use the list of such graphs given in the literature to verify existing results on equations with three, four and five variables and to prove new results for equations with six variables. We start with 14 nonisomorphic graphs with ten vertices, choose a set of 14 representative parastrophically nonequivalent equations and give their general solutions. A case of equations with seven and more variables is briefly discussed. The problem of Sokhats'kyi concerning a property which distinguishes visually two parastrophically nonequivalent equations with four variables is solved.