Modular knight distance in graphs and applications on the [formula omitted]-queens problem

Abstract

In this paper we introduce modular knight distance on a chessboard and use it to study the geometry of a class of solutions of the (modular) n-queens problem. Such geometry is related to the movement of a knight in classic chess, therefore we benefit from defining modular knight distance prior to the study. The concept of the distance is incorporated to graph theory with the notion of the knight graph, from which we obtain several properties which allows us to study the desired class of solutions. In the end of the study we obtain an algorithm with relatively reduced complexity which can be used to generate all the solutions of the class. The idea of such algorithm can be used for computing all the solutions of broader classes. Additionally we prove a result about the analytical classification of all the solutions of the class in interest.