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Planar graphs without 5-cycles and intersecting triangles are $$(1, 1, 0)$$-colorable. (English) Zbl 1327.05117
Summary: A $$(c_1, c_2, \ldots, c_k)$$-coloring of $$G$$ is a mapping $$\varphi : V(G) \mapsto \{1, 2, \ldots, k \}$$ such that for every $$i$$, $$1 \leq i \leq k$$, $$G [V_i]$$ has maximum degree at most $$c_i$$, where $$G [V_i]$$ denotes the subgraph induced by the vertices colored $$i$$. O. V. Borodin and A. Raspaud [J. Comb. Theory, Ser. B 88, No. 1, 17–27 (2003; Zbl 1023.05046)] conjecture that every planar graph without 5-cycles and intersecting triangles is $$(0, 0, 0)$$-colorable. We prove in this paper that such graphs are $$(1, 1, 0)$$-colorable.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C10 Planar graphs; geometric and topological aspects of graph theory
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##### References:
 [1] Borodin, O. V., Colorings of plane graphs: a survey, Discrete Math., 313, 517-539, (2013) · Zbl 1259.05042 [2] Borodin, O. V.; Glebov, A. N., A sufficient condition for planar graphs to be 3-colorable, Diskret Anal Issled Oper., 10, 3-11, (2004), (in Russian) · Zbl 1045.05041 [3] Borodin, O. V.; Glebov, A. N., Planar graphs with neither 5-cycles nor close 3-cycles are 3-colorable, J. Graph Theory, 66, 1-31, (2011) · Zbl 1237.05067 [4] Borodin, O. V.; Glebov, A. N.; Raspaud, A. R.; 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 [5] 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 [6] G. Chang, F. Havet, M. Montassier, A. Raspaud, Steinberg’s Conjecture and near colorings, manuscript. [7] Z. Dvoˇrák, D. Král, R. Thomas, Coloring planar graphs with triangles far apart, Mathematics ArXiV, arXiv:0911.0885, 2009. [8] Grötzsch, H., Ein dreifarbensatz für dreikreisfreienetze auf der kugel, Math.-Nat.Reihe, 8, 109-120, (1959) [9] Havel, I., On a conjecture of grunbaum, J. Combin. Theory, Ser. B, 7, 184-186, (1969) · Zbl 0177.26805 [10] Hill, O.; Smith, D.; Wang, Y.; Xu, L.; Yu, G., Planar graphs without 4-cycles and 5-cycles are $$(3, 0, 0)$$-colorable, Discrete Math., 313, 2312-2317, (2013) · Zbl 1281.05055 [11] Hill, O.; Yu, G., A relaxation of steinberg’s conjecture, SIAM J. Discrete Math., 27, 584-596, (2013) · Zbl 1268.05074 [12] Liu, R.; Li, X.; Yu, G., A relaxation of the Bordeaux conjecture, European J. Combin., 49, 240-249, (2015) · Zbl 1315.05059 [13] Steinberg, R., The state of the three color problem. quo vadis, graph theory?, Ann. Discrete Math., 55, 211-248, (1993) [14] Xu, B., A 3-color theorem on plane graph without 5-circuits, Acta Math. Sin., 23, 1059-1062, (2007) · Zbl 1122.05038 [15] Xu, B., On $$(3, 1)^\ast$$-coloring of planar graphs, SIAM J. Discrete Math., 23, 205-220, (2008) [16] Xu, L.; Miao, Z.; Wang, Y., Every planar graph with cycles of length neither 4 nor 5 is $$(1, 1, 0)$$-colorable, J. Comb. Optim., 28, 4, 774-786, (2014) · Zbl 1309.05058 [17] Xu, L.; Wang, Y., Improper colorability of planar graphs with cycles of length neither 4 nor 6 (in Chinese), Sci Sin Math, 43, 15-24, (2013) [18] C. Yang, C. Yerger, The Bordeaux 3-color Conjecture and Near-Coloring, preprint.
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