# Leaper graphs

@inproceedings{Knuth1994LeaperG,
title={Leaper graphs},
author={Donald Ervin Knuth},
year={1994}
}
• D. Knuth
• Published 1 November 1994
• Mathematics
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 Science
FUN
• 2021
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
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.
Pohl-Warnsdorf – Revisited
• Mathematics
• 2004
Two new series of graphs are introduced. Properties of these graphs make them suitable as test beds for Hamiltonian path heuristics. The graphs are planar and regular of degree 3, and each series has
Combinatorial Games: selected Bibliography with a Succinct Gourmet Introduction
The family of combinatorial games consists of two-player games with perfect information, no hidden information as in some card games, no chance moves and outcome restricted to (lose, win), (tie, tie) and (draw, draw) for the two players who move alternately.
Knight’s Tours on Rectangular Chessboards Using External Squares
• Mathematics
• 2014
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
Reachability in Restricted Walk on Integers
• Mathematics
J. Univers. Comput. Sci.
• 2010
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
The Closed Knight Tour Problem in Higher Dimensions
• Mathematics
Electron. J. Comb.
• 2012
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?
• Mathematics
Electron. J. Comb.
• 2011
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.
Tours of a generalized knight on rectangular chessboards
• São Paulo Journal of Mathematical Sciences
• 2022

## References

SHOWING 1-10 OF 10 REFERENCES
A guide to fairy chess
A current branching circuit for an amplifier for first and second input terminals and a current supply feed terminal comprising a pair of magnetically coupled windings with one of the windings
The five free leapers
• The five free leapers
• 1976
Traité des Applications de l'Analyse Mathématique au Jeu des Echecs
• Traité des Applications de l'Analyse Mathématique au Jeu des Echecs
Theory of leapers
• Theory of leapers
• 1985
Solution d'une question curieuse qui ne paroit soumisè a aucune analyse
• Mémoires de l'Academie Royale des Sciences et Belles Letters
Generalized knights and Hamiltonian tours
• Generalized knights and Hamiltonian tours
• 1993
Fairy Chess Review
• Fairy Chess Review
• 1945
Letter to the editor, The British Chess Magazine
• Letter to the editor, The British Chess Magazine
• 1918