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On 3-colorable plane graphs without 5- and 7-cycles. (English) Zbl 1108.05046
Summary: In this note, it is proved that every plane graph without 5- and 7-cycles and without adjacent triangles is 3-colorable. This improves the result of O. V. Borodin, A. N. Glebov, A. Raspaud and M. R. Salavatipour [J. Comb. Theory, Ser. B 93, 303–311 (2005; Zbl 1056.05052)], and offers a partial solution for a conjecture of O. V. Borodin and A. Raspaud [J. Comb. Theory, Ser. B 88, 17–27 (2003; Zbl 1023.05046)].

MSC:
05C15 Coloring of graphs and hypergraphs
05C38 Paths and cycles
Keywords:
coloring
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References:
[1] 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
[2] Borodin, O.V.; Raspaud, A., A sufficient condition for planar graphs to be 3-colorable, J. combin. theory ser. B, 88, 17-27, (2003) · Zbl 1023.05046
[3] Steinberg, R., The state of the three color problem, (), 211-248 · Zbl 0791.05044
[4] B. Xu, A 3-color theorem on plane graphs without 5-circuits, Acta Math. Sinica, in press · Zbl 1122.05038
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