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Defective 2-colorings of planar graphs without 4-cycles and 5-cycles. (English) Zbl 1388.05072
Summary: A 2-coloring is a coloring of vertices of a graph with colors 1 and 2. Define $$V_i := \{v \in V(G) : c(v) = i \}$$ for $$i = 1$$ and 2. We say that $$G$$ is $$(d_1, d_2)$$-colorable if $$G$$ has a 2-coloring such that $$V_i$$ is an empty set or the induced subgraph $$G [V_i]$$ has the maximum degree at most $$d_i$$ for $$i = 1$$ and 2. Let $$G$$ be a planar graph without 4-cycles and 5-cycles. We show that the problem to determine whether $$G$$ is $$(0, k)$$-colorable is NP-complete for every positive integer $$k$$. Moreover, we construct non-$$(1, k)$$-colorable planar graphs without 4-cycles and 5-cycles for every positive integer $$k$$. In contrast, we prove that $$G$$ is $$(d_1, d_2)$$-colorable where $$(d_1, d_2) = (4, 4),(3, 5),$$ and $$(2, 9)$$.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C10 Planar graphs; geometric and topological aspects of graph theory 05C38 Paths and cycles 68Q25 Analysis of algorithms and problem complexity
##### Keywords:
defective coloring; planar graph; cycle
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##### References:
  Appel, K.; Haken, W., Every planar map is four colorable. I. discharging, Illinois J. Math., 21, 3, 429-490, (1977) · Zbl 0387.05009  Appel, K.; Haken, W.; Koch, J., Every planar map is four colorable. II. reducibility, Illinois J. Math., 21, 3, 491-561, (1977) · Zbl 0387.05010  Borodin, O. V.; Ivanova, A. O.; Montassier, M.; Ochem, P.; Raspaud, A., Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most $$k$$, J. Graph Theory, 65, 83-93, (2010) · Zbl 1209.05177  Borodin, O. V.; Kostochka, A. V., Defective 2-colorings of sparse graphs, J. Combin. Theory Ser. B, 104, 72-80, (2014) · Zbl 1282.05041  Chen, M.; Wang, Y.; Liu, P.; Xu, J., Planar graphs without cycles of length $$4$$ or $$5$$ are $$(2, 0, 0)$$-colorable, Discrete Math., 339, 661-667, (2016)  Choi, H.; Choi, I.; Jeong, J.; Suh, G., $$(1, k)$$-coloring of graphs with girth at least $$5$$ on a surface, J. Graph Theory, 84, 4, 521-535, (2017) · Zbl 1359.05099  Choi, I.; Raspaud, A., Planar graphs with minimum cycle length at least $$5$$ are $$(3, 5)$$-colorable, Discrete Math., 338, 4, 661-667, (2015) · Zbl 1305.05072  Cohen-Addad, V.; Hebdige, M.; Král, D.; Li, Z.; Salgado, E., Steinberg’s conjecture is false, J. Combin. Theory Ser. B, (2016), Available Online 26 July · Zbl 1350.05018  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  Eaton, N.; Hull, T., Defective List colorings of planar graphs, Bull. Inst. Combin. Appl., 25, 9-87, (1999) · Zbl 0916.05026  Grötzsch, H., Zur theorie der diskreten gebilde. VII ein dreifarbensatz für dreikreisfreie netze auf der kugel, (Wiss. Z Martin-Luther-Univ. Halle-Wittenberg, Math.-Nat. Reihe, vol. 8, (1958/1959)), 109-120, (in German)  Havet, F.; Sereni, J.-S., Improper choosability of graphs and maximum average degree, J. Graph Theory, 52, 181-199, (2006) · Zbl 1104.05026  Montassier, M.; Ochem, P., Near-colorings: non-colorable graphs and NP-completeness, Electron. J. Combin., 22, 1, 13, (2015), Paper 157 · Zbl 1308.05052  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
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