快乐边：正则格的Happy边着色
Happy Edges: Threshold-Coloring of Regular Lattices
摘要：
我们研究了一个有趣的数独风格的难题所激发的图着色问题。给定一个图的边分为{ \em near}和{ \em far}两个集合以及一个整数阈值$t$，图的{ \em阈值着色}是对顶点的整数赋值，使得近边的端点相差$t$或更小，而远边的端点相差大于$t$。我们研究阈值染色的平铺平面的正多边形，被称为阿基米德格子，和他们的兄弟，Laves格子。我们证明了一些是阈值着色与任何边缘标签的颜色数量恒定，一些需要一个无界的颜色数量为特定的标签，和一些不是阈值着色。
英文摘要：
We study a graph coloring problem motivated by a fun Sudoku-style puzzle. Given a bipartition of the edges of a graph into {\em near} and {\em far} sets and an integer threshold $t$, a {\em threshold-coloring} of the graph is an assignment of integers to the vertices so that endpoints of near edges differ by $t$ or less, while endpoints of far edges differ by more than $t$. We study threshold-coloring of tilings of the plane by regular polygons, known as Archimedean lattices, and their duals, the Laves lattices. We prove that some are threshold-colorable with constant number of colors for any edge labeling, some require an unbounded number of colors for specific labelings, and some are not threshold-colorable.
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