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Abstract: A sequence $r_1,r_2,\dots,r_{2n}$ such that $r_i=r_{n+i}$ for all $1\leq i \leq n$, is called a {\em repetition}. A sequence $S$ is called {\em non-repetitive} if no subsequence of consecutive terms of $S$ is a repetition. Let $G$ be a graph whose edges are coloured. A trail in $G$ is called {\em non-repetitive} if the sequence of colours of its edges is non-repetitive. If $G$ is a plane graph, a {\em facial non-repetitive edge-colouring} of $G$ is an edge-colouring such that any {\it facial trail} is non-repetitive. We denote $\pi'_f(G)$ the minimum number of colours needed. In this paper we prove that for graphs of Platonic, Archimedean and prismatic polyhedra $\pi'_f(G)$ is either $3$ or $4$. Contact the authors: stanislav.jendrol@upjs.sk, erika.skrabulakova@upjs.sk Download PDF version of the preprint. |