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Relaxed Two-Coloring of Cubic Graphs

Abstract : 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 induces monochromatic components of order at most $189$. 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.
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Robert Berke, Tibor Szabó. Relaxed Two-Coloring of Cubic Graphs. 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), 2005, Berlin, Germany. pp.341-344. ⟨hal-01184434⟩

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