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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
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