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Relaxed two-coloring of cubic graphs. (English) Zbl 1118.05029
Summary: We show that any graph of maximum degree at most 3 has a two-coloring such that one color-class is an independent set, while the other color-class induces monochromatic components of order at most 750. On the other hand, for any constant $$C$$, we exhibit a 4-regular graph such that the deletion of any independent set leaves at least one component of order greater than $$C$$. Similar results are obtained for coloring graphs of given maximum degree with $$k+\ell$$ colors such that $$k$$ parts form an independent set and $$\ell$$ parts span components of order bounded by a constant. A lot of interesting questions remain open.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
##### Keywords:
bounded degree graphs; colorings
Full Text:
##### References:
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