# An improper planar coloring with monochromatic component size $\leq 2$

Let us relax the coloring a little bit, that is, we allow a small number of adjacent vertices to be assigned the same color. A monochromatic component is defined to be a connected component in the subgraph induced by the set of vertices that receive the same color, and the question is to ask for the minimum number $\lambda$ of colors needed to color a graph such that largest monochromatic component has size no more than $C$.
The traditional coloring can be considered as $[\lambda,1]$-coloring in this setting. Hence to find the minimum number of $\lambda$ is NP-hard for planar graph in general.

My question is, how about $[\lambda,2]$-coloring of planar graphs, or more generally, $[\lambda,C]$-coloring for $C \geq 2$?

This can be viewed as a dual problem of what is studied by Edwards and Farr, where $\lambda$ is fixed, and one is asked to find the minimum size of $C$.

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