# A note on cyclic chromatic number

Discussiones Mathematicae Graph Theory (2010)

- Volume: 30, Issue: 1, page 115-122
- ISSN: 2083-5892

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topJana Zlámalová. "A note on cyclic chromatic number." Discussiones Mathematicae Graph Theory 30.1 (2010): 115-122. <http://eudml.org/doc/270892>.

@article{JanaZlámalová2010,

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.},

author = {Jana Zlámalová},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {plane graph; cyclic colouring; cyclic chromatic number},

language = {eng},

number = {1},

pages = {115-122},

title = {A note on cyclic chromatic number},

url = {http://eudml.org/doc/270892},

volume = {30},

year = {2010},

}

TY - JOUR

AU - Jana Zlámalová

TI - A note on cyclic chromatic number

JO - Discussiones Mathematicae Graph Theory

PY - 2010

VL - 30

IS - 1

SP - 115

EP - 122

AB - 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.

LA - eng

KW - plane graph; cyclic colouring; cyclic chromatic number

UR - http://eudml.org/doc/270892

ER -

## References

top- [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. Zbl0611.05037
- [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. Zbl0565.05027
- [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. Zbl1190.05052
- [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. Zbl1185.05060
- [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. Zbl0655.05030
- [9] H. Whitney, Congruent graphs and the connectivity of graphs, Amer. J. Math. 54 (1932) 150-168, doi: 10.2307/2371086. Zbl58.0609.01

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