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

Voir la notice de l'article provenant de la source Library of Science

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 
@article{DMGT_2010_30_1_a9,
     author = {Zl\'amalov\'a, Jana},
     title = {A note on cyclic chromatic number},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {115--122},
     publisher = {mathdoc},
     volume = {30},
     number = {1},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2010_30_1_a9/}
}
TY  - JOUR
AU  - Zlámalová, Jana
TI  - A note on cyclic chromatic number
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2010
SP  - 115
EP  - 122
VL  - 30
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2010_30_1_a9/
LA  - en
ID  - DMGT_2010_30_1_a9
ER  - 
%0 Journal Article
%A Zlámalová, Jana
%T A note on cyclic chromatic number
%J Discussiones Mathematicae. Graph Theory
%D 2010
%P 115-122
%V 30
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2010_30_1_a9/
%G en
%F DMGT_2010_30_1_a9
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/

[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)30:3177::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.