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Planar graphs without 4, 6, 8-cycles are 3-colorable. (English) Zbl 1144.05033
It was conjectured by Steinberg in 1976 that every planar graph without 4-cycles and 5-cycles is 3-colorable [see T. Jensen and B. Toft, Graph Coloring Problems, New York, NY: John Wiley & Sons (1995; Zbl 0855.05054)] and it was asked by Erdős whether there exists an integer \(k\) such that the absence of cycles with length from 4 to \(k\) in a planar graph guarantees its 3-colorability. In this paper the authors prove that every planar graph without cycles of length 4, 6, and 8 is 3-colorable.

MSC:
05C15 Coloring of graphs and hypergraphs
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[1] Jensen R T, Toft B. Graph Coloring Problems. New York: Wiley, 1995 · Zbl 0855.05054
[2] Grötzsch H. Ein Drefarbensatz fur dreikreisfreie Netze auf der Kugel. Wiss Z Martin-Luther-Univ Halle-Wittenberg, Mat-Natur Reche, 8: 109–120 (1959)
[3] Steinberg R. The state of the three color problem. In: Gimbel J, Kenndy J W, Quintas L V, eds. Quo Vadis, Graph Theory? Annals of Discrete Math, Vol 55, 1993, 211–248 · Zbl 0791.05044
[4] Abbott H L, Zhou B. On small faces in 4-critical graphs. Ars Combin, 32: 203–207 (1991) · Zbl 0755.05062
[5] Borodin O V. Structural properties of plane graphs without adjacent triangles and an application to 3-colorings. J Graph Theory, 21: 183–186 (1996) · Zbl 0838.05039 · doi:10.1002/(SICI)1097-0118(199602)21:2<183::AID-JGT7>3.0.CO;2-N
[6] Sanders D P, Zhao Y. A note on the three coloring problem. Graphs Combin, 11: 92–94 (1995) · Zbl 0824.05023 · doi:10.1007/BF01787424
[7] Borodin O V, Glebov A N, Raspaud A, Salavatipour M R. Planar graphs without cycles of length from 4 to 7 are 3-colorable. J Combin Theory Ser B, 93: 303–311 (2005) · Zbl 1056.05052 · doi:10.1016/j.jctb.2004.11.001
[8] Xu B. On 3-colorable plane graphs without 5-and 7-cycles. J Combin Theory Ser B, 96(6): 958–963 (2006) · Zbl 1108.05046 · doi:10.1016/j.jctb.2006.02.005
[9] Zhang L, Wu B. A note on 3-choosability of planar graphs without certain cycles. Discrete Math, 297: 206–209 (2005) · Zbl 1070.05046 · doi:10.1016/j.disc.2005.05.001
[10] Chen M, Raspaud A, Wang W. Three-coloring planar graphs without short cycles. Inform Process Lett, 101: 134–138 (2007) · Zbl 1185.05057 · doi:10.1016/j.ipl.2006.08.009
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