To cite this version:O.V. Borodin, A.O. Ivanova, Mickael Montassier, Pascal Ochem, André Raspaud. Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k. Journal of Graph Theory, Wiley, 2010, 65 (2) is (k, 0)-colorable. In particular, it follows that every planar graph with girth at least 7 is (8, 0)-colorable. On the other hand, we construct planar graphs with girth 6 that are not (k, 0)-colorable for arbitrarily large k.
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