论文标题
整数晶格的一般位置数
The general position number of integer lattices
论文作者
论文摘要
连接图$ g $的一般职位编号$ {\ rm gp}（g）$是最大的$ s $顶点的基数，因此，$ s $的三个成对截然不同的顶点均来自$ s $，躺在普通的大地测量上。 $ n $二维网格图$ \ pn $是两条无限路径$ p_ \ p_ \ infty $ of $ n $副本的笛卡尔产品。证明，如果$ n \ in {\ mathbb n} $，则$ {\ rm gp}（{p_ \ infty^n}）= 2^{2^{2^{n-1}} $。结果较早以$ n \ in \ {1,2 \} $而闻名，部分以$ n = 3 $而闻名。
The general position number ${\rm gp}(G)$ of a connected graph $G$ is the cardinality of a largest set $S$ of vertices such that no three pairwise distinct vertices from $S$ lie on a common geodesic. The $n$-dimensional grid graph $\pn$ is the Cartesian product of $n$ copies of the two-way infinite path $P_\infty$. It is proved that if $n\in {\mathbb N}$, then ${\rm gp}({P_\infty^n}) = 2^{2^{n-1}}$. The result was earlier known only for $n\in \{1,2\}$ and partially for $n=3$.
