图的极限非周期和重复染色
Limit aperiodic and repetitive colorings of graphs
摘要：
设$X$是一个顶点度上界$Δ$有限的（重复）无限连通单图.主要定理指出，$X$允许一个由$Δ$颜色进行的（重复）极限非周期顶点染色.这完善了由Collins和Trenk，以及Klavžar，Wong和Zhu独立证明的有限图定理。它也与Gao，Jackson和Seward的一个定理有关，该定理指出任何可数群都有两种颜色的强非周期染色，以及Lehner，Pilganniak和Stawiski以及Hüning等人最近对图的可区分数的研究。我们得出类似的结果，边缘着色，更一般的图形，以及建设有限的非周期性和重复的瓷砖由许多prototiles。在随后的文件中，这一结果也被用来改善建设的紧凑叶空间与规定的叶子。
英文摘要：
Let $X$ be a (repetitive) infinite connected simple graph with a finite upper bound $Δ$ on the vertex degrees. The main theorem states that $X$ admits a (repetitive) limit aperiodic vertex coloring by $Δ$ colors. This refines a theorem for finite graphs proved by Collins and Trenk, and by Klavžar, Wong and Zhu, independently. It is also related to a theorem of Gao, Jackson and Seward stating that any countable group has a strongly aperiodic coloring by two colors, and to recent research on distinguishing number of graphs by Lehner, Pilśniak and Stawiski, and by Hüning et al. In our theorem, the number of colors is optimal for general graphs of bounded degree. We derive similar results for edge colorings, and for more general graphs, as well as a construction of limit aperiodic and repetitive tilings by finitely many prototiles. In a subsequent paper, this result is also used to improve the construction of compact foliated spaces with a prescribed leaf.
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