论文标题
方格彩色问题
A square-grid coloring problem
论文作者
论文摘要
假设$ n \ ge 2 $，我们希望在$ n \ times n $ square网格的正方形中种植$ k $不同类型的树木。我们可以随心所欲地拥有每种类型的数量。唯一的规则是，每对类型都必须发生在网格中某个地方的相邻正方形中。问题是：给定$ n $，$ k $最大的是什么？用$γ（n）$表示这个数字，并将其称为$ n \ times n $网格的完整着色号码 *。一点想法表明$γ（n）\ le 2n-1 $。我们感兴趣的主要问题是每$ n \ ge 2 $ $γ（n）= 2n-1 $。
Suppose that $n \ge 2$, and we wish to plant $k$ different types of trees in the squares of an $n \times n$ square grid. We can have as many of each type as we want. The only rule is that every pair of types must occur in an adjacent pair of squares somewhere in the grid. The question is: given $n$, what is the largest that $k$ can be? Denote this number by $Γ(n)$, and call this the *complete coloring number* of the $n \times n$ grid. A little thought shows that $Γ(n) \le 2n-1$. The main question we are interested in is whether $Γ(n) = 2n-1$ for every $n \ge 2$.
