Jordan区域和曲线的着色
Coloring Jordan regions and curves
摘要：
若当区域是平面的一个子集，它同胚于一个闭圆盘。考虑一族$\mathcal{F}$ Jordan区域，其内部是两两不相交的，并且使得任何两个Jordan区域至多相交于一点。如果任何一点的平面中包含在最多$k$元素的$\mathcal{F}$（$k$足够大），那么我们表明，元素的$\mathcal{F}$可以着色与最多$k+1$的颜色，使交叉约旦地区被分配不同的颜色。这是最好的可能性，并回答了Reed和Shepherd在1996年提出的问题。作为一个简单的推论，我们还得到了Hlinhnnanjin（1998）关于弦的接触系的色数的一个问题的肯定答案.我们还研究了接触Jordan曲线族的色数。这可以用来限制平面有向图中顶点不相交的有向圈的最大数目与其分数对应部分之间的比率。
英文摘要：
A Jordan region is a subset of the plane that is homeomorphic to a closed disk. Consider a family $\mathcal{F}$ of Jordan regions whose interiors are pairwise disjoint, and such that any two Jordan regions intersect in at most one point. If any point of the plane is contained in at most $k$ elements of $\mathcal{F}$ (with $k$ sufficiently large), then we show that the elements of $\mathcal{F}$ can be colored with at most $k+1$ colors so that intersecting Jordan regions are assigned distinct colors. This is best possible and answers a question raised by Reed and Shepherd in 1996. As a simple corollary, we also obtain a positive answer to a problem of Hliněný (1998) on the chromatic number of contact systems of strings. We also investigate the chromatic number of families of touching Jordan curves. This can be used to bound the ratio between the maximum number of vertex-disjoint directed cycles in a planar digraph, and its fractional counterpart.
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