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… 
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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
Les carrés magiques
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