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Every planar graph without 4-cycles and 5-cycles is $$(2, 6)$$-colorable. (English) Zbl 1437.05077
Summary: A graph is $$(d_1,\ldots ,d_r)$$-colorable if the vertex set can be partitioned into $$r$$ sets $$V_1,\ldots ,V_r$$ where the maximum degree of the subgraph induced by $$V_i$$ is at most $$d_i$$ for each $$i\in \{1,\ldots ,r\}$$. In this paper, we prove that every planar graph without 4-cycles and 5-cycles is $$(2, 6)$$-colorable, which improves the result of P. Sittitrai and K. Nakprasit [Discrete Math. 341, No. 8, 2142–2150 (2018; Zbl 1388.05072)], who proved that every planar graph without 4-cycles and 5-cycles is $$(2, 9)$$-colorable.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C10 Planar graphs; geometric and topological aspects of graph theory 05C38 Paths and cycles
##### Keywords:
improper coloring; planar graph; discharging method
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##### References:
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