• DOI:10.2307/3620202
• Corpus ID: 16856513

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 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. Theorem 1. The graph of an {r, s}-leaper on an m × n board, when 2 ≤ m ≤ n and 1 ≤ r ≤ s, is connected if and only if the following three conditions hold: (i) r + s is relatively prime to r − s; (ii) n ≥ 2s; (iii) m ≥ r + s. Proof. Condition (i) is necessary because any common divisor d of r + s and r − s will be a divisor of x + y for any vertex (x, y) reachable from (0, 0); any leaper move changes the sum of coordinates by ±(r + s) or ±(s − r). If d > 1, vertex (0, 1) would therefore be disconnected from (0, 0). Condition (ii) is necessary because the " middle " point (⌊m/2⌋, ⌊n/2⌋) would otherwise be isolated: If m ≤ n < 2s we have ⌊m/2⌋ − s < 0, ⌊m/2⌋ + s ≥ m, ⌊n/2⌋ − s < 0, and ⌊n/2⌋ + s ≥ n, so there is no place to leap from there. To show that condition (iii) is necessary we show first that the {r, s}-leaper graph on any board with m = r + s − 1 and n = ∞ has no path from (0, 0) to (0, 1). For this purpose we construct a special path through points (x k , y k) as follows: The path terminates when x k = s − 1. It is not difficult to see that termination will occur when …