Discrete Mathematics & Theoretical Computer Science 
We show that any graph of maximum degree at most $3$ has a twocoloring, such that one colorclass 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.
Source : ScholeXplorer
IsRelatedTo ARXIV 1705.06928 Source : ScholeXplorer IsRelatedTo DOI 10.1002/jgt.22462 Source : ScholeXplorer IsRelatedTo DOI 10.48550/arxiv.1705.06928
