Latin Hexahedra and Related Combinatorial Structures

Abstract

AbstractWe introduce a system of Latin rectangles that is a combination of finite number of Latin rectangles that are given by either concrete Latin rectangles or variables representing a Latin rectangle. Using such systems, we prove the existence of a combinatorial object which is considered as a generalization of a Latin square. It is defined on the surface of a regular hexahedron so that any subarray of any net of the hexahedron is a Latin rectangle. Second, we introduce two combinatorial objects, the existence of which are equivalent to 1-factorizations of the complete tripartite graph \(K_{2n,2n,2n}\) and the complete quadripartite graph \(K_{n,n,n,n}\), respectively. We also show how to construct 1-factorizations of \(K_{2n,2n,2n}\) and \(K_{n,n,n,n}\).KeywordsLatin squareRegular hexahedronSudoku puzzle1-factorizationComplete tripartite graphComplete quadripartite graph