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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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