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Abstract: We consider a vertex colouring of a connected plane graph $G$. A colour $c$ is used $k$ times by a face $\alpha$ of $G$ if it appears $k$ times along the facial walk of $\alpha$. We prove that every connected plane graph with minimum face degree at least 3 has a vertex colouring with four colours such that every face uses some colour an odd number of times. We conjecture that such a colouring can be done using three colours. We prove that this conjecture is true for 2-connected cubic plane graphs. Next we consider other three kinds of colourings that require stronger restrictions. Contact the authors: julius.czap@upjs.sk, stanislav.jendrol@upjs.sk Download PDF version of the preprint. |