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A new upper bound on the cyclic chromatic number

By O. V. Borodin, H. J. Broersma, A. Glebov and Jan van den Heuvel

Abstract

A cyclic colouring of a plane graph is a vertex colouring such that vertices incident with the same face have distinct colours. The minimum number of colours in a cyclic colouring of a graph is its cyclic chromatic number Âc. Let ¢¤ be the maximum face degree of a graph. There exist plane graphs with Âc = b3 2 ¢¤c. Ore and Plummer (1969) proved that Âc · 2¢¤, which bound was improved to b9 5 ¢¤c by Borodin, Sanders and Zhao (1999), and to d5 3 ¢¤e by Sanders and Zhao (2001). 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 Âc · max{¢¤ + 3 k¤ + 2, ¢¤ + 14, 3 k¤ + 6, 18 }, and if ¢¤ ¸ 4 and k¤ ¸ 4, then Âc · ¢¤ + 3 k¤ + 2

Topics: QA Mathematics
Publisher: Centre for Discrete and Applicable Mathematics, London School of Economics and Political Science
Year: 2004
OAI identifier: oai:eprints.lse.ac.uk:13346
Provided by: LSE Research Online
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