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Planar graphs with girth at least 5 are $$(3, 5)$$-colorable. (English) Zbl 1305.05072
Summary: A graph is $$(d_1, \ldots, d_r)$$-colorable if its vertex set can be partitioned into $$r$$ sets $$V_1, \ldots, V_r$$ where the maximum degree of the graph induced by $$V_i$$ is at most $$d_i$$ for each $$i \in \{1, \ldots, r \}$$. Let $$\mathcal{G}_g$$ denote the class of planar graphs with minimum cycle length at least $$g$$. We focus on graphs in $$\mathcal{G}_5$$ since for any $$d_1$$ and $$d_2$$, M. Montassier and P. Ochem [“Near-colorings: non-colorable graphs and NP-completeness”, submitted] constructed graphs in $$\mathcal{G}_4$$ that are not $$(d_1, d_2)$$-colorable. It is known that graphs in $$\mathcal{G}_5$$ are $$(2, 6)$$-colorable and $$(4, 4)$$-colorable, but not all of them are $$(3, 1)$$-colorable. We prove that graphs in $$\mathcal{G}_5$$ are $$(3, 5)$$-colorable, leaving two interesting questions open: (1) Are graphs in $$\mathcal{G}_5$$ also $$(3, d_2)$$-colorable for some $$d_2 \in \{2, 3, 4 \}$$? (2) Are graphs in $$\mathcal{G}_5$$ indeed $$(d_1, d_2)$$-colorable for all $$d_1 + d_2 \geq 8$$ where $$d_2 \geq d_1 \geq 1$$?

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