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A graph is $(d_1, \ldots, d_k)$-colorable if its vertex set can bepartitioned into $k$ nonempty subsets so that the subgraph induced by the $i$thpart has maximum degree at most $d_i$ for each $i\in\{1, \ldots, k\}$. It isknown 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 findingthe 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 graphswith 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 aforementionedresult 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.

Planar graphs with girth at least 5 are (3,4)-colorable