Geodesic
Near-proper vertex 2-colorings of sparse graphs
Diskretnyj analiz i issledovanie operacij, Tome 16 (2009) no. 2, pp. 16-20.

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A graph $G$ is $(2,1)$-colorable if its vertices can be partitioned into subsets $V_1$ and $V_2$ such that in $G[V_1]$ any component contains at most two vertices while $G[V_2]$ is edgeless. We prove that every graph $G$ with the maximum average degree $\mathrm{mad}(G)$ smaller than 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 $\mathrm{mad}(G_n)=(18n-2)/(7n-1)$ that is not $(2,1)$-colorable. Bibl. 5.
Keywords: planar graph, girth, coloring
Mots-clés : partition. 
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O. V. Borodin; A. O. Ivanova. Near-proper vertex 2-colorings of sparse graphs. Diskretnyj analiz i issledovanie operacij, Tome 16 (2009) no. 2, pp. 16-20. http://geodesic.mathdoc.fr/item/DA_2009_16_2_a1/

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