论文标题
飞机上的着色距离图
Coloring distance graphs on the plane
论文作者
论文摘要
我们考虑欧几里得平面上某些距离图的着色。也就是说，我们要求以使平面所有点为所有点上色所需的颜色数量最小，以使得在间隔$ [1，b] $中的距离对成对的点成对。经典的Hadwiger-Nelson问题是这个问题的特殊情况 - 通过服用$ b = 1 $获得。本文的主要结果改善了$ b $的颜色数量的下限和上限。特别是，我们确定了$ b $的两个值范围的颜色数量最少 - 其中一个是扩大了Exoo提出的间隔，第二个是全新的。据我们所知，这些是平面上唯一已知的距离图，具有确定的非平凡色数。此外，我们提出了$ b $的第一个$ 8 $颜色，大于$ b $的$ b $，用于已知的$ 7 $颜色。作为副产品，我们给出了平面边界部分的一些界限和精确值，特别是通过涂色。
We consider the coloring of certain distance graphs on the Euclidean plane. Namely, we ask for the minimal number of colors needed to color all points of the plane in such a way that pairs of points at distance in the interval $[1,b]$ get different colors. The classic Hadwiger-Nelson problem is a special case of this question -- obtained by taking $b=1$. The main results of the paper are improved lower and upper bounds on the number of colors for some values of $b$. In particular, we determine the minimal number of colors for two ranges of values of $b$ - one of which is enlarging an interval presented by Exoo and the second is completely new. Up to our knowledge, these are the only known families of distance graphs on the plane with a determined nontrivial chromatic number. Moreover, we present the first $8$-coloring for $b$ larger than values of $b$ for the known $7$-colorings. As a byproduct, we give some bounds and exact values for bounded parts of the plane, specifically by coloring certain annuli.
