A note on cyclic chromatic number

• Autorzy: Jana Zlámalová
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

plane graph; cyclic colouring; cyclic chromatic number

Kategorie tematyczne

• 05C15: Coloring of graphs and hypergraphs

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2010
• Tom: 30
• Numer: 1
• Strony: 115-122

Daty

wydano

2010

otrzymano

2008-06-04

poprawiono

2009-04-16

zaakceptowano

2009-04-16

Twórcy

autor

Jana Zlámalová

• Institute of Mathematics, Faculty of Science, P.J. Šafárik University, Jesenná 5, 040 01 Košice, Slovakia

Bibliografia

• [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.

Identyfikatory

DOI

10.7151/dmgt.1481