  In this work, we continue the study of vertex colorings of graphs, in which
adjacent vertices are allowed to be of the same color as long as each
monochromatic connected component is of relatively small cardinality. We focus
on colorings with two and three available colors and present improved bounds on
the size of the monochromatic connected components for two meaningful
subclasses of planar graphs, namely maximal outerplanar graphs and complete
planar 3-trees.
