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Discussiones Mathematicae Graph Theory 30(1) (2010) 115-122
DOI: https://doi.org/10.7151/dmgt.1481

A NOTE ON CYCLIC CHROMATIC NUMBER

Jana Zlámalová

Institute of Mathematics, Faculty of Science
P.J. Safárik University
Jesenná 5, 040 01 Košice, Slovakia
e-mail: zlamalovaj@gmail.com

Abstract

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.

2010 Mathematics Subject Classification: 05C15.

References

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Received 4 June 2008
Revised 16 April 2009
Accepted 16 April 2009