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A 3-color theorem on plane graphs without 5-circuits. (English) Zbl 1122.05038
Summary: We prove that every plane graph without 5-circuits and without triangles of distance less than 3 is 3-colorable. This improves the main result of O. V. Borodin and A. Raspaud [J. Comb. Theory, Ser. B 88, 17–27 (2003; Zbl 1023.05046)], and provides a new upper bound to their conjecture.

MSC:
05C15 Coloring of graphs and hypergraphs
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[1] Steinberg, R.: The state of the three color problem, Quo Vadis. Graph Theory? J. Gimbel, J. W. Kennedy & L. V. Quintas (eds). Ann Discrete Math., 55, 211–248 (1993) · Zbl 0791.05044
[2] Borodin, O. V.: Structural properties of plane graphs without adjacent triangles and an application to 3-colorings. Journal of Graph Theory, 21(2), 183–186 (1996) · Zbl 0838.05039
[3] Abbott, H. L., Zhou, B.: On small faces in 4-critical graphs. Ars Combin., 32, 203–207 (1991) · Zbl 0755.05062
[4] Sanders, D. P., Zhao, Y.: A note on the three color problem. Graphs and Combinatorics, 11, 91–94 (1995) · Zbl 0824.05023
[5] Borodin, O. V., et al.: Planar graphs without cycles of length from 4 to 7 are 3-colorable. Journal of Combinatorial Theory, Ser. B, 93, 303–311 (2005) · Zbl 1056.05052
[6] Xu, B.: On 3-colorable plane graphs without 5- and 7-circuits. Journal of Combinatorial Theory, Ser. B, in press · Zbl 1108.05046
[7] Borodin, O. V., Raspaud, A.: A su.cient condition for planar graphs to be 3-colorable. Journal of Combinatorial Theory, Ser. B, 88, 17–27 (2003) · Zbl 1023.05046
[8] Xu, B.: On 3-colorings of plane graphs. Acta Math. Appl. Sinica, 20, 597–604 (2004) · Zbl 1062.05063
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