Hamiltonian decomposition of K^∗_n, patterns with distinct differences, and Tuscan squares

Abstract

This paper presents a few constructions for the decomposition of the complete directed graph on n vertices into n Hamiltonian paths. Some of the constructions will apply for even n and others to odd n. The constructions will be obtained from some patterns with distinct differences. The constructions will be exhibited by squares (called Tuscan squares) which sometimes are Latin squares (called Roman squares), and sometimes are not Latin. These squares have some special properties which are discussed in this paper.