Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k

Abstract : A graph $G$ is $(k,0)$-colorable if its vertices can be partitioned into subsets $V_1$ and $V_2$ such that in $G[V_1]$ every vertex has degree at most $k$, while $G[V_2]$ is edgeless. For every integer $k\ge 1$, we prove that every graph with the maximum average degree smaller than $\frac {3k+4}{k+2}$ 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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Submitted on : Tuesday, April 21, 2009 - 4:05:02 PM
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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), pp.83-93. ⟨hal-00377372⟩

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