Leaper graphs
Abstract
An $\{r,s\}$leaper is a generalized knight that can jump from $(x,y)$ to $(x\pm r,y\pm s)$ or $(x\pm s,y\pm r)$ on a rectangular grid. The graph of an $\{r,s\}$leaper on an $m\times n$ board is the set of $mn$~vertices $(x,y)$ for $0\leq x<m$ and $0\leq y<n$, with an edge between vertices that are one $\{r,s\}$leaper move apart. We call $x$ the {\it rank} and $y$ the {\it 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 $r=1$.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 1994
 arXiv:
 arXiv:math/9411240
 Bibcode:
 1994math.....11240K
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 Math. Gazette 78 (1994), 274297