2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
Stefan Felsner (ed.)
DMTCS Conference Volume AE (2005), pp. 341344
author:  Robert Berke and Tibor Szabó 

title:  Relaxed TwoColoring of Cubic Graphs 
keywords:  Vertex coloring, bounded size components 
abstract: 
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+ℓ
colors such that
k
parts form an independent set and
ℓ
parts span components of order bounded by a constant.
A lot of interesting questions remain open.

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reference:  Robert Berke and Tibor Szabó (2005), Relaxed TwoColoring of Cubic Graphs, in 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), Stefan Felsner (ed.), Discrete Mathematics and Theoretical Computer Science Proceedings AE, pp. 341344 
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