跳跃图
Leaper graphs
摘要：
$\{r,s\}$ -跳跃者是一个广义的骑士，可以在矩形网格上从$(x,y)$跳到$(x\pm r,y\pm s)$或$(x\pm s,y\pm r)$。$m\times n$板上的$\{r,s\}$ -跳跃器的图是$0\leq x<m$和$0\leq y<n$的$mn$ ~顶点$(x,y)$的集合，其中顶点之间的边是一个$\{r,s\}$ -跳跃器移动分开。我们称$x$为{ \it等级}，$y$为棋盘位置$(x,y)$的{ \it文件}。George~ P. Jelliss提出了几个关于这些图的有趣问题，并建立了它们的一些基本性质。本文的目的是对任意的~ $r$和~ $s$，刻画图连通时的特征，并确定当$s=r+1$或$r=1$时具有哈密顿回路的最小电路板.
英文摘要：
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 $r=1$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
