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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
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