Abstract
A graph G is (2, 1)-colorable if its vertices can be partitioned into subsets V 1 and V 2 such that each component in G[V 1] contains at most two vertices while G[V 2] is edgeless. We prove that every graph with maximum average degree mad(G) < 7/3 is (2, 1)-colorable. It follows that every planar graph with girth at least 14 is (2, 1)-colorable. We also construct a planar graph G n with mad (G n ) = (18n − 2)/(7n − 1) that is not (2, 1)-colorable.
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Original Russian Text © O.V. Borodin, A.O. Ivanova, 2009, published in Diskretnyi Analiz i Issledovanie Operatsii, 2009, Vol. 16, No. 2, pp. 16–20.
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Borodin, O.V., Ivanova, A.O. Near-proper vertex 2-colorings of sparse graphs. J. Appl. Ind. Math. 4, 21–23 (2010). https://doi.org/10.1134/S1990478910010047
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DOI: https://doi.org/10.1134/S1990478910010047