Defective choosability of graphs in surfaces

Douglas R. Woodall

Discussiones Mathematicae Graph Theory, Tome 31 (2011), p. 441-459 / Harvested from The Polish Digital Mathematics Library

Résumé

It is known that if G is a graph that can be drawn without edges crossing in a surface with Euler characteristic ε, and k and d are positive integers such that k ≥ 3 and d is sufficiently large in terms of k and ε, then G is (k,d)*-colorable; that is, the vertices of G can be colored with k colors so that each vertex has at most d neighbors with the same color as itself. In this paper, the known lower bound on d that suffices for this is reduced, and an analogous result is proved for list colorings (choosability). Also, the recent result of Cushing and Kierstead, that every planar graph is (4,1)*-choosable, is extended to $K_{3,3}$-minor-free and K₅-minor-free graphs.

Détail

Publié le : 2011-01-01

Zbl 1229.05107

EUDML-ID : urn:eudml:doc:270878

Bibliographie

[000] [1] D. Archdeacon, A note on defective colorings of graphs in surfaces, J. Graph Theory 11 (1987) 517-519, doi: 10.1002/jgt.3190110408. | Zbl 0654.05030

[001] [2] L.J. Cowen, R.H. Cowen and D.R. Woodall, Defective colorings of graphs in surfaces: partitions into subgraphs of bounded valency, J. Graph Theory 10 (1986) 187-195, doi: 10.1002/jgt.3190100207. | Zbl 0596.05024

[002] [3] L. Cowen, W. Goddard and C.E. Jesurum, Defective coloring revisited, J. Graph Theory 24 (1997) 205-219, doi: 10.1002/(SICI)1097-0118(199703)24:3<205::AID-JGT2>3.0.CO;2-T | Zbl 0877.05019

[003] [4] W. Cushing and H.A. Kierstead, Planar graphs are 1-relaxed, 4-choosable, European J. Combin. 31 (2010) 1385-1397, doi: 10.1016/j.ejc.2009.11.013. | Zbl 1221.05077

[004] [5] G.A. Dirac, A property of 4-chromatic graphs and some remarks on critical graphs, J. London Math. Soc. 27 (1952) 85-92, doi: 10.1112/jlms/s1-27.1.85. | Zbl 0046.41001

[005] [6] N. Eaton and T. Hull, Defective list colorings of planar graphs, Bull Inst. Combin. Appl. 25 (1999) 79-87. | Zbl 0916.05026

[006] [7] S. Gutner, The complexity of planar graph choosability, Discrete Math. 159 (1996) 119-130, doi: 10.1016/0012-365X(95)00104-5. | Zbl 0865.05066

[007] [8] W. He, W. Miao and Y. Shen, Another proof of the 5-choosability of K₅-minor-free graphs, Discrete Math. 308 (2008) 4024-4026, doi: 10.1016/j.disc.2007.07.089. | Zbl 1155.05023

[008] [9] M. Mirzakhani, A small non-4-choosable planar graph, Bull Inst. Combin. Appl. 17 (1996) 15-18. | Zbl 0860.05029

[009] [10] N. Robertson, P.D. Seymour and R. Thomas, Hadwiger's conjecture for K₆-free graphs, Combinatorica 13 (1993) 279-361, doi: 10.1007/BF01202354. | Zbl 0830.05028

[010] [11] R. Skrekovski, Choosability of K₅-minor-free graphs, Discrete Math. 190 (1998) 223-226, doi: 10.1016/S0012-365X(98)00158-7. | Zbl 0955.05043

[011] [12] R. Skrekovski, List improper colourings of planar graphs, Combin. Probab. Comput. 8 (1999) 293-299, doi: 10.1017/S0963548399003752. | Zbl 0940.05031

[012] [13] C. Thomassen, Every planar graph is 5-choosable, J. Combin. Theory (B) 62 (1994) 180-181, doi: 10.1006/jctb.1994.1062. | Zbl 0805.05023

[013] [14] M. Voigt, List colourings of planar graphs, Discrete Math. 120 (1993) 215-219, doi: 10.1016/0012-365X(93)90579-I. | Zbl 0790.05030

[014] [15] K. Wagner, Über eine Eigenschaft der ebenen Komplexe, Math. Ann. 114 (1937) 570-590, doi: 10.1007/BF01594196. | Zbl 63.0550.01

[015] [16] D.R. Woodall, Improper colourings of graphs, in: R. Nelson and R.J. Wilson (eds.), Graph Colourings, Pitman Research Notes in Math. 218 (Longman, Harlow, Essex, 1990) 45-63. | Zbl 0693.05028

[016] [17] D.R. Woodall, List Colourings of Graphs, in: J.W.P. Hirschfeld (ed), Surveys in Combinatorics, 2001, London Math. Soc. Lecture Note Series 288 (Cambridge University Press, 2001) 269-301.