A new upper bound on the cyclic chromatic number

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 χc . Let ∆∗ be the maximum face degree of a graph. There exist plane graphs with χc = 3 2 ∆∗ . Ore and Plummer (1969) proved that χc ≤ 2 ∆∗ , which bound was improved to 9 5 ∆∗ by Borodin, Sanders and Zhao (1999), and to 5 3 ∆∗ 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 χc ≤ max{ ∆∗ + 3 k∗ + 2, ∆∗ + 14, 3 k∗ + 6, 18 }, and if ∆∗ ≥ 4 and k∗ ≥ 4, then χc ≤ ∆∗ + 3 k∗ + 2.

• Type: Journal
• Year: 2007
• Where: JGT
• Authors: Oleg V. Borodin, Hajo Broersma, Alexei N. Glebov, J. van den Heuvel
• Keywords: Cyclic Chromatic Number, Cyclic Colouring, JGT 2007, Plane Graphs