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Planar graphs without cycles of length 4 or 5 are \((2, 0, 0)\)-colorable. (English) Zbl 1327.05073
Summary: Let \(d_1, d_2, \ldots, d_k\) be \(k\) nonnegative integers. A graph \(G = (V, E)\) is \((d_1, d_2, \ldots, d_k)\)-colorable, if the vertex set \(V\) of \(G\) can be partitioned into subsets \(V_1, V_2, \ldots, V_k\) such that the subgraph \(G [V_i]\) induced by \(V_i\) has maximum degree at most \(d_i\) for \(i = 1, 2, \ldots, k\). R. Steinberg et al. [Holland. Ann. Discrete Math. 55, 211–248 (1993; Zbl 0791.05044)] conjectured that planar graphs without cycles of length 4 or 5 are \((0, 0, 0)\)-colorable. O. Hill et al. [Discrete Math. 313, No. 20, 2312–2317 (2013; Zbl 1281.05055)] showed that every planar graph without cycles of length 4 or 5 is \((3, 0, 0)\)-colorable. In this paper, we show that planar graphs without cycles of length 4 or 5 are \((2, 0, 0)\)-colorable. For further study in this direction, some problems and conjectures are presented.

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
05C15 Coloring of graphs and hypergraphs
05C38 Paths and cycles
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[1] Abbott, H. L.; Zhou, B., On small faces in 4-critical graphs, Ars Combin., 32, 203-207, (1991) · Zbl 0755.05062
[2] Appel, K.; Haken, W., Every planar map is four colorable, part I. discharging, Illinois J. Math., 21, 429-490, (1997) · Zbl 0387.05009
[3] Appel, K.; Haken, W., Every planar map is four colorable, part II. reducibility, Illinois J. Math., 21, 491-567, (1977) · Zbl 0387.05010
[4] Bondy, J. A.; Murty, U. S.R., Graph theory, (2008), Springer Berlin · Zbl 1134.05001
[5] Borodin, O. V., Structural properties of plane graphs without adjacent triangles and an application to 3-colorings, J. Graph Theory, 21, 2, 183-186, (1996) · Zbl 0838.05039
[6] 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
[7] Bu, Y.; Xu, J.; Wang, Y., \((1, 0, 0)\)-colorability of planar graphs without prescribed short cycles, J. Comb. Optim., 30, 627-646, (2015) · Zbl 1331.90061
[8] G. Chang, F. Havet, M. Montassier, A. Raspaud, Steinberg’s Conjecture and Near Colorings, http://hal.inria.fr/inria-00605810/en/.
[9] Cowen, L. J.; Cowen, R. H.; Woodall, D. R., Defective colorings of graphs in surfaces: partitions into subgraphs of bounded valency, J. Graph Theory, 10, 187-195, (1986) · Zbl 0596.05024
[10] Grötzsch, H., Ein dreifarbensatz fur dreikreisfreie netze auf der kugel, wiss Z martin luther univ halle-Wittenberg, Math-nat, 8, 109-120, (1959)
[11] Hill, O.; Smith, D.; Wang, Y.; Xu, L.; Yu, G., Planar graphs without 4-cycles or 5-cycles are \((3, 0, 0)\)-colorable, Discrete Math., 313, 2312-2317, (2013) · Zbl 1281.05055
[12] Hill, O.; Yu, G., A relaxation of steinberg’s conjecture, SIAM J. Discrete Math., 27, 1, 584-596, (2013) · Zbl 1268.05074
[13] Li, H.; Xu, J.; Wang, Y., Planar graphs with cycles of length neither 4 nor 7 are \((3, 0, 0)\)-colorable, Discrete Math., 327, 29-35, (2014) · Zbl 1288.05094
[14] Lih, K.; Song, Z.; Wang, W.; Zhang, K., A note on List improper coloring planar graphs, Appl. Math. Lett., 14, 269-273, (2001) · Zbl 0978.05029
[15] Liu, P. P.; Wang, Y. Q., Planar graphs without cycles of length 4 or 7 are \((2, 0, 0)\)-colorable, Sci. Sin. Math., 44, 1153-1164, (2014), (in Chinese)
[16] Sanders, D. P.; Zhao, Y., A note on the three coloring problem, Graphs Combin., 11, 91-94, (1995) · Zbl 0824.05023
[17] Steinberg, R., The state of the three color problem, (Gimbel, J.; Kenndy, J. W.; Quintas, L. V.; Vadis, Quo, Graph Theory, Ann. Diserete Math., 55, (1993)), 211-248 · Zbl 0791.05044
[18] Wang, Y.; Jin, L.; Kang, Y., Planar graphs without cycles of length from 4 to 6 are \((1, 0, 0)\)-colorable, Sci. Sin. Math., 43, 1145-1164, (2013), (in Chinese)
[19] Wang, Y.; Xu, L., Improper choosability of planar graphs without 4-cycles, SIAM J. Discrete Math., 27, 4, 2029-2037, (2013) · Zbl 1291.05077
[20] Wang, Y.; Xu, J., Planar graphs with cycles of length neither 4 nor 6 are \((2, 0, 0)\)-colorable, Inform. Process. Lett., 113, 659-663, (2013) · Zbl 1285.05071
[21] Wang, Y.; Xu, J., Improper colorability of planar graphs without prescribed short cycles, Discrete Math., 322, 5-14, (2014) · Zbl 1283.05114
[22] Wang, Y.; Xu, J., Decomposing a planar graph without cycles of length 5 into a matching and a 3-colorable graph, European J. Combin., 43, 98-123, (2015) · Zbl 1301.05273
[23] Wang, Y.; Yang, Y., \((1, 0, 0)\)-colorability of planar graphs with cycles of length 4, 5 or 9, Discrete Math., 326, 44-49, (2014) · Zbl 1288.05105
[24] Xu, B., On \((3, 1)^\ast\)-coloring of plane graphs, SIAM J. Discrete Math., 23, 1, 205-220, (2009)
[25] 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, 774-786, (2014) · Zbl 1309.05058
[26] Xu, L.; Wang, Y., Improper colorability of planar graphs with cycles of length neither 4 nor 6, Sci. Sin. Math., 43, 1, 15-24, (2013), (in Chinese)
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