Discrete Mathematics & Theoretical Computer Science 
A vertex coloring of a graph $G$ is $k \textit{nonrepetitive}$ if one cannot find a periodic sequence with $k$ blocks on any simple path of $G$. The minimum number of colors needed for such coloring is denoted by $\pi _k(G)$ . This idea combines graph colorings with Thue sequences introduced at the beginning of 20th century. In particular Thue proved that if $G$ is a simple path of any length greater than $4$ then $\pi _2(G)=3$ and $\pi_3(G)=2$. We investigate $\pi_k(G)$ for other classes of graphs. Particularly interesting open problem is to decide if there is, possibly huge, $k$ such that $\pi_k(G)$ is bounded for planar graphs.
Source : ScholeXplorer
IsRelatedTo ARXIV 1904.05269 Source : ScholeXplorer IsRelatedTo DOI 10.19086/aic.12100 Source : ScholeXplorer IsRelatedTo DOI 10.48550/arxiv.1904.05269
