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Leaper graphs

Published online by Cambridge University Press:  01 August 2016

Donald E. Knuth
Donald E. Knuth*
Affiliation:
Computer Science Dept, Stanford University, Stanford CA 94305, USA
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Extract

An {r, s}-leaper 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 x 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 raised several interesting questions about these graphs, and established some of their fundamental properties. The purpose of this paper is to characterize when the graphs are connected, for arbitrary r and s, and to determine the smallest boards with Hamiltonian circuits when s = r + 1. or r = 1. (A Hamiltonian circuit is a closed path that visits every cell exactly once.)

Type
Research Article
Information
The Mathematical Gazette , Volume 78 , Issue 483 , November 1994 , pp. 274 - 297
Copyright
Copyright © The Mathematical Association 1994

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