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Abstract: A cyclic colouring of a graph $G$ embedded in a surface is a vertex colouring of $G$ in which any two distinct vertices sharing a face receive distinct colours. The cyclic chromatic number $\chic(G)$ of $G$ is the smallest number of colours in a cyclic colouring of $G$. Plummer and Toft in~1987 conjectured that $\chic(G)\le\Delta^*+2$ for any 3-connected plane graph $G$ with maximum face degree $\Delta^*$. It is known that the conjecture holds true for $\Delta^*\le4$ and $\Delta^*\ge18$. The validity of the conjecture is proved in the paper for some special classes of planar graphs. Contact the authors: jana.zlamalova@upjs.sk Download PDF version of the preprint. |