Geodesic
A note on cyclic chromatic number
Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 1, pp. 115-122

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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 χ_c(G) of G is the smallest number of colours in a cyclic colouring of G. Plummer and Toft in 1987 conjectured that χ_c(G) ≤ Δ* + 2 for any 3-connected plane graph G with maximum face degree Δ*. It is known that the conjecture holds true for Δ* ≤ 4 and Δ* ≥ 18. The validity of the conjecture is proved in the paper for some special classes of planar graphs.
Keywords: plane graph, cyclic colouring, cyclic chromatic number 
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Zlámalová, Jana. A note on cyclic chromatic number. Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 1, pp. 115-122. http://geodesic.mathdoc.fr/item/DMGT_2010_30_1_a9/