Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Which Chessboards have a Closed Knight’s Tour within the Cube. | Which Chessboards have a Closed Knight s Tour within the Cube Joe DeMaio Department of Mathematics and Statistics Kennesaw State University Kennesaw Georgia 30144 USA jdemaio@ Submitted Feb 28 2007 Accepted Apr 26 2007 Published May 9 2007 Mathematics Subject Classification 05C45 00A08 Abstract A closed knight s tour of a chessboard uses legal moves of the knight to visit every square exactly once and return to its starting position. When the chessboard is translated into graph theoretic terms the question is transformed into the existence of a Hamiltonian cycle. There are two common tours to consider on the cube. One is to tour the six exterior n X n boards that form the cube. The other is to tour within the n stacked copies of the n X n board that form the cube. This paper is concerned with the latter. In this paper necessary and sufficient conditions for the existence of a closed knight s tour for the cube are proven. 1 Introduction The closed knight s tour of a chessboard is a classic problem in mathematics. Can the knight use legal moves to visit every square on the board and return to its starting position The unique movement of the knight makes its tour an intriguing problem which is trivial for other chess pieces. The knight s tour is an early example of the existence problem of Hamiltonian cycles. So early in fact that it predates Kirkman s 1 1856 paper which posed the general problem and Hamilton s Icosian Game of the late 1850s 2 . Euler presented solutions for the standard 8 X 8 board 3 and the problem is easily generalized to rectangular boards. In 1991 Schwenk 4 completely answered the question Which rectangular chessboards have a knight s tour Schwenk s Theorem An m X n chessboard with m n has a closed knight s tour unless one or more of the following three conditions hold a m and n are both odd b m 2 1 2 4g c m 3 and n 2 4 6 8g. THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R32 1 The problem of the closed knight s tour has been further
