The Closed Knight Tour Problem in Higher Dimensions
نویسندگان
چکیده
The problem of existence of closed knight tours for rectangular chessboards was solved by Schwenk in 1991. Last year, in 2011, DeMaio and Mathew provide an extension of this result for 3-dimensional rectangular boards. In this article, we give the solution for n-dimensional rectangular boards, for n > 4.
منابع مشابه
Magic Knight's Tours in Higher Dimensions
A knight‟s tour on a board is a sequence of knight moves that visits each square exactly once. A knight‟s tour on a square board is called magic knight‟s tour if the sum of the numbers in each row and column is the same (magic constant). Knight‟s tour in higher dimensions (n > 3) is a new topic in the age-old world of knight‟s tours. In this paper, it has been proved that there can‟t be magic k...
متن کاملWhich Chessboards have a Closed Knight's Tour within the Rectangular Prism?
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. In 1991 Schwenk completely classified the m × n rectangular chessboards that admit a closed knight’s tour. In honor of the upcoming twentieth anniversary of the publication of Schwenk’s paper, this article extends his result by classifying the i × j × k r...
متن کاملComplexity and Stop Conditions for NP as General Assignment Problems, the Travel Salesman Problem in R2, Knight Tour Problem and Boolean Satisfiability Problem
This paper presents stop conditions for solving General Assignment Problems (GAP), in particular for Travel Salesman Problem in an Euclidian 2D space the well known condition Jordan’s simple curve and opposite condition for the Knight Tour Problem. The Jordan’s simple curve condition means that a optimal trajectory must be simple curve, i.e., without crossing but for Knight Tour Problem we use ...
متن کاملWhich Chessboards have a Closed Knight's Tour within the Cube?
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 × n boards that form the cube. The ot...
متن کاملGeneralised Knight's Tours
The problem of existence of closed knight’s tours in [n]d, where [n] = {0, 1, 2, . . . , n− 1}, was recently solved by Erde, Golénia, and Golénia. They raised the same question for a generalised, (a, b) knight, which is allowed to move along any two axes of [n]d by a and b unit lengths respectively. Given an even number a, we show that the [n]d grid admits an (a, 1) knight’s tour for sufficient...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Electr. J. Comb.
دوره 19 شماره
صفحات -
تاریخ انتشار 2012