Vertex coloring of plane graphs with nonrepetitive boundary paths

A sequence $s_1,s_2,\mathrm{.}\mathrm{.}\mathrm{.},s_k,s_1,s_2,\mathrm{.}\mathrm{.}\mathrm{.},s_k$ is a repetition. A sequence $S$ is nonrepetitive, if no subsequence of consecutive terms of $S$ form a repetition. Let $G$ be a vertex colored graph. A path of $G$ is nonrepetitive, if the sequence of colors on its vertices is nonrepetitive. If $G$ is a plane graph, then a facial nonrepetitive vertex coloring of $G$ is a vertex coloring such that any facial path is nonrepetitive. Let ${\pi }_f(G)$ denote the minimum number of colors of a facial nonrepetitive vertex coloring of $G$. Jendro\vl and Harant posed a conjecture that ${\pi }_f(G)$ can be bounded from above by a constant. We prove that ${\pi }_f(G)\le 24$ for any plane graph $G$.