In this paper we prove that every planar graph without 4, 5 and 8-cycles is 3-colorable.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1579, author = {Sakib A. Mondal}, title = {Planar graphs without 4-, 5- and 8-cycles are 3-colorable}, journal = {Discussiones Mathematicae Graph Theory}, volume = {31}, year = {2011}, pages = {775-789}, zbl = {1255.05083}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1579} }
Sakib A. Mondal. Planar graphs without 4-, 5- and 8-cycles are 3-colorable. Discussiones Mathematicae Graph Theory, Tome 31 (2011) pp. 775-789. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1579/
[000] [1] H.L. Abbott and B. Zhou, On small faces in 4-critical graphs, Ars Combin. 32 (1991) 203-207. | Zbl 0755.05062
[001] [2] O.V. Borodin, Structural properties of plane graphs without adjacent triangles and an application to 3-colorings, J. Graph Theory 21 (1996 ) 183-186, doi: 10.1002/(SICI)1097-0118(199602)21:2<183::AID-JGT7>3.0.CO;2-N | Zbl 0838.05039
[002] [3] O.V. Borodin, To the paper: 'On small faces in 4-critical graphs', Ars Combin. 43 (1996) 191-192. | Zbl 0881.05037
[003] [4] O.V. Borodin, A.N. Glebov, A. Raspaud and M.R. Salavatipour, Planar graphs without cycles of length from 4 to 7 are 3-colorable, J. Combin. Theory (B) 93 (2005) 303-311, doi: 10.1016/j.jctb.2004.11.001. | Zbl 1056.05052
[004] [5] O.V. Borodin and A.N. Glebov, A sufficient condition for plane graphs to be 3-colorable, Diskret Analyz Issled. Oper. 10 (2004) 3-11.
[005] [6] O.V. Borodin and A. Raspaud, A sufficient condition for planar graph to be 3-colorable, J. Combin. Theory (B) 88 (2003) 17-27, doi: 10.1016/S0095-8956(03)00025-X. | Zbl 1023.05046
[006] [7] O.V. Borodin, A.N. Glebov, M. Montassier and A. Raspaud, Planar graphs without 5- and 7-cycles and without adjacent triangles are 3-colorable, J. Combin. Theory (B) 99 (2009) 668-673, doi: 10.1016/j.jctb.2008.11.001. | Zbl 1184.05024
[007] [8] M. Chen, A. Raspaud and W. Wang, Three-coloring planar graphs without short cycles, Information Processing Letters 101 (2007) 134-138, doi: 10.1016/j.ipl.2006.08.009. | Zbl 1185.05057
[008] [9] H. Grotsch, Ein Dreifarbensatz fur dreikreisfreie Netze auf der Kugel, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg, Mat.-Natur. Reihe 8 (1959) 102-120.
[009] [10] I. Havel, On a conjecture of Grünbaum, J. Combin. Theory (B) 7 (1969), 184-186, doi: 10.1016/S0021-9800(69)80054-2. | Zbl 0177.26805
[010] [11] D.P. Sanders and Y. Zhao, A note on the three color problem, Graphs and Combin. 11 (1995) 91-94, doi: 10.1007/BF01787424. | Zbl 0824.05023
[011] [12] R. Steinberg, The state of the three color problem, in: Ouo Vadis, Graph Theory? 55 (1993) 211-248. | Zbl 0791.05044
[012] [13] W. Wang. and M. Chen, Planar graphs without 4,6,8-cycles are 3-colorable, Science in China Series A: Mathematics (Science in China Press, co-published with Springer-Verlag GmbH) 50 (2007) 1552-1562. | Zbl 1144.05033
[013] [14] L. Xiaofang, M. Chen and W. Wang, On 3-colorable planar graphs without cycles of four lengths, Information Processing Letters 103 (2007) 150-156, doi: 10.1016/j.ipl.2007.03.007. | Zbl 1185.05061
[014] [15] B.Xu, A 3-color theorem on plane graph without 5-circuits, Acta Mathematica Sinica 23 (2007) 1059-1062, doi: 10.1007/s10114-005-0851-7. | Zbl 1122.05038
[015] [16] L. Zhang and B. Wu, A note on 3-choosability of planar graphs without certain cycles, Discrete Math. 297 (2005) 206-209, doi: 10.1016/j.disc.2005.05.001. | Zbl 1070.05046