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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: [email protected]
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 χc(G) of G is the smallest number of colours in a cyclic colouring of G. Plummer and Toft in 1987 conjectured that χc(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
| [1] | K. Ando, H. Enomoto and A. Saito, Contractible edges in 3-connected graphs, J. Combin. Theory (B) 42 (1987) 87-93, doi: 10.1016/0095-8956(87)90065-7. |
| [2] | O.V. Borodin, Solution of Ringel's problem on vertex-face coloring of plane graphs and coloring of 1-planar graphs (Russian), Met. Diskr. Anal. 41 (1984) 12-26. |
| [3] | H. Enomoto and M. Hornák, A general upper bound for the cyclic chromatic number of 3-connected plane graphs, J. Graph Theory 62 (2009) 1-25, doi: 10.1002/jgt.20383. |
| [4] | H. Enomoto, M. Hornák and S. Jendrol', Cyclic chromatic number of 3-connected plane graphs, SIAM J. Discrete Math. 14 (2001) 121-137, doi: 10.1137/S0895480198346150. |
| [5] | M. Hornák and S. Jendrol', On a conjecture by Plummer and Toft, J. Graph Theory 30 (1999) 177-189, doi: 10.1002/(SICI)1097-0118(199903)30:3<177::AID-JGT3>3.0.CO;2-K. |
| [6] | M. Hornák and J. Zlámalová, Another step towards proving a conjecture by Plummer and Toft, Discrete Math. 310 (2010) 442-452, doi: 10.1016/j.disc.2009.03.016. |
| [7] | A. Morita, Cyclic chromatic number of 3-connected plane graphs (Japanese, M.S. Thesis), Keio University, Yokohama 1998. |
| [8] | M.D. Plummer and B. Toft, Cyclic coloration of 3-polytopes, J. Graph Theory 11 (1987) 507-515, doi: 10.1002/jgt.3190110407. |
| [9] | H. Whitney, Congruent graphs and the connectivity of graphs, Amer. J. Math. 54 (1932) 150-168, doi: 10.2307/2371086. |
Received 4 June 2008
Revised 16 April 2009
Accepted 16 April 2009
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