3 problems

Let $G$ be a plane graph, and let $\pi_f(G)$ denote its facial nonrepetitive chromatic number, the minimum number of colours in a vertex-colouring such that every facial boundary w…

Let $G$ be a planar graph, and let $\pi(G)$ denote its nonrepetitive chromatic number, the minimum number of colours required to colour the vertices of $G$ so that every path has a…

Let $P$ be a path, and let $\pi_T(P)$ denote its total Thue chromatic number, the minimum number of colours in a total colouring of $P$ with no repetitive sequence of consecutive v…