Limit aperiodic and repetitive colorings of graphs

Abstract

Let $X$ be a (repetitive) infinite connected simple graph with a finite upper bound $\Delta$ on the vertex degrees. The main theorem states that $X$ admits a (repetitive) limit aperiodic vertex coloring by $\Delta$ colors. This refines a theorem for finite graphs proved by Collins and Trenk, and by Klav\v{z}ar, 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, Pil\'{s}niak and Stawiski, and by H\"{u}ning 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.