Facial Nonrepetitive Colorings

of Biconnected Outerplanar Graphs

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).