Facial Nonrepetitive Colorings
of Biconnected Outerplanar Graphs
Lucas Rioux Maldague

For a plane graph $G$, the facial nonrepetitive coloring number, denoted $\pi_{f(G)}$, is the minimum number of colors required to color the vertices of $G$ such that any path that lies on the vertices around a single face of $G$ is nonrepetitive. A path is nonrepetitive if the sequence of colors of consecutive vertices of the path cannot be written as $aa$, where $a$ is a sequence of colors. We show that $\pi_{f(G)} \leq 7$ for any biconnected outerplanar graph $G$. This improves on the best known bound of 12 by Kündgen and Pelsmajer (2008).