A new upper bound on the cyclic chromatic number


Borodin, O.V. and Broersma, H.J. and Glebov, A. and Heuvel, J. van den (2006) A new upper bound on the cyclic chromatic number. Journal of Graph Theory, 54 (1). pp. 58-72. ISSN 0364-9024

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Abstract:A cyclic coloring of a plane graph is a vertex coloring such that vertices incident with the same face have distinct colors. The minimum number of colors in a cyclic coloring of a graph is its cyclic chromatic number $\chi^c$. Let $\Delta^*$ be the maximum face degree of a graph. There exist plane graphs with $\chi^c = \lfloor {3\over 2}\Delta^*\rfloor$. Ore and Plummer [5] proved that $\chi^c \le 2\Delta^*$, which bound was improved to $\lfloor {9\over 5}\Delta^*\rfloor$ by Borodin, Sanders, and Zhao [1], and to $\lceil {5\over 3}\Delta^*\rceil$ by Sanders and Zhao [7].

We introduce a new parameter $k^*$, which is the maximum number of vertices that two faces of a graph can have in common, and prove that $\chi^c \le \max \{\Delta^* + 3k^* + 2,\Delta^* + 14, 3k^* + 6, 18\}$, and if $\Delta^* \ge  4$ and $k^* \ge 4$, then $\chi^c\le \Delta^* + 3k^* + 2$.
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Electrical Engineering, Mathematics and Computer Science (EEMCS)
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Link to this item:http://purl.utwente.nl/publications/63866
Official URL:https://doi.org/10.1002/jgt.20193
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