Leaper graphs
@inproceedings{Knuth1994LeaperG, title={Leaper graphs}, author={Donald Ervin Knuth}, year={1994} }
An {r, s}-leaper [1, p. 130; 2, p. 30; 3] is a generalized knight that can jump from (x, y) to (x±r, y±s) or (x ± s, y ± r) on a rectangular grid. The graph of an {r, s}-leaper on an m × n board is the set of mn vertices (x, y) for 0 ≤ x < m and 0 ≤ y < n, with an edge between vertices that are one {r, s}-leaper move apart. We call x the rank and y the file of board position (x, y). George P. Jelliss [4, 5] raised several interesting questions about these graphs, and established some of their…
10 Citations
Taming the Knight's Tour: Minimizing Turns and Crossings
- Computer ScienceFUN
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The techniques are generalized to rectangular boards, high-dimensional boards, symmetric tours, odd boards with a missing corner, and tours for $(1,4)$-leapers, and it is shown that these extensions also admit a constant approximation ratio on the minimum number of turns, and on the number of crossings in most cases.
Knight's Tours in Higher Dimensions
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It is shown that on the d-dimensional board with n even, there is always a knight's tour provided that n is sufficiently large, and an exact classification of the grids in which there is a knight’s tour is given.
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Knight’s Tours on Rectangular Chessboards Using External Squares
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The classic puzzle of finding a closed knight’s tour on a chessboard consists of moving a knight from square to square in such a way that it lands on every square once and returns to its starting…
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We prove that two conditions are sucient, and with three exceptions also necessary, for reachability of any position in restricted walk on integers in which the sizes of the moves to the left and to…
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The solution of existence of closed knight tours for rectangular chessboards for rectangular boards for n-dimensional rectangular boards is given.
Which Chessboards have a Closed Knight's Tour within the Rectangular Prism?
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In honor of the upcoming twentieth anniversary of the publication of Schwenk's paper, this article extends his result by classifying thei\times j\times k$ rectangular prisms that admit a closed knight's tour.
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Tours of a generalized knight on rectangular chessboards
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