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Planar graphs with girth at least 5 are $$(3, 4)$$-colorable. (English) Zbl 1422.05035
Summary: A graph is $$(d_1, \ldots, d_k)$$-colorable if its vertex set can be partitioned into $$k$$ nonempty subsets so that the subgraph induced by the $$i$$-th part has maximum degree at most $$d_i$$ for each $$i \in \{1, \ldots, k \}$$. It is known that for each pair $$(d_1, d_2)$$, there exists a planar graph with girth $$4$$ that is not $$(d_1, d_2)$$-colorable. This sparked the interest in finding the pairs $$(d_1, d_2)$$ such that planar graphs with girth at least $$5$$ are $$(d_1, d_2)$$-colorable. Given $$d_1 \leq d_2$$, it is known that planar graphs with girth at least $$5$$ are $$(d_1, d_2)$$-colorable if either $$d_1 \geq 2$$ and $$d_1 + d_2 \geq 8$$ or $$d_1 = 1$$ and $$d_2 \geq 10$$. We improve an aforementioned result by providing the first pair $$(d_1, d_2)$$ in the literature satisfying $$d_1 + d_2 \leq 7$$ where planar graphs with girth at least $$5$$ are $$(d_1, d_2)$$-colorable. Namely, we prove that planar graphs with girth at least $$5$$ are $$(3, 4)$$-colorable.

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