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2-colorable perfect matching in cubic planar graphs is very similar to your problem which was stated to be NP-complete by Schaefer in his famous dichotomy theorem paper although he did not give the proof for cubic planar graphs. The problem asks for the existence of two coloring of cubic planar graphs such that every vertex has exactly one neighbor of the same color as itself.

EDIT: Defective coloring is the decision version of your problem. A graph is (k, d)-colorable if one can color the vertices with k colors such that no vertex is adjacent to more than d vertices of its same color. The decision problem (2,1)-coloring which is equivalent to your problem was shown to be NP-complete even for planar graphs.

2-colorable perfect matching in cubic planar graphs is very similar to your problem which was stated to be NP-complete by Schaefer in his famous dichotomy theorem paper although he did not give the proof for cubic planar graphs. The problem asks for the existence of two coloring of cubic planar graphs such that every vertex has exactly one neighbor of the same color as itself.

EDIT: Defective coloring is the decision version of your problem. A graph is (k, d)-colorable if one can color the vertices with k colors such that no vertex is adjacent to more than d vertices of its same color. The decision problem (2,1)-coloring with defects, which is equivalent to your optimization problem, was shown to be NP-complete even for planar graphs.

2-colorable perfect matching in cubic planar graphs is very similar to your problem which was stated to be NP-complete by Schaefer in his famous dichotomy theorem paper although he did not give the proof for cubic planar graphs. The problem asks for the existence of two coloring of cubic planar graphs such that every vertex has exactly one neighbor of the same color as itself.

2-colorable perfect matching in cubic planar graphs is very similar to your problem which was stated to be NP-complete by Schaefer in his famous dichotomy theorem paper although he did not give the proof for cubic planar graphs. The problem asks for the existence of two coloring of cubic planar graphs such that every vertex has exactly one neighbor of the same color as itself.

EDIT: Defective coloring is the decision version of your problem. A graph is (k, d)-colorable if one can color the vertices with k colors such that no vertex is adjacent to more than d vertices of its same color. The decision problem (2,1)-coloring which is equivalent to your problem was shown to be NP-complete even for planar graphs.

2-colorable perfect matching in cubic planar graphs is reducible to your problem which was stated to be NP-complete by Schaefer in his famous dichotomy theorem paper although he did not give the proof for cubic planar graphs. The problem asks for the existence of two coloring of cubic planar graphs such that every vertex has exactly one neighbor of the same color as itself.

Therefore, your problem is NP-complete when $C$ =2.

2-colorable perfect matching in cubic planar graphs is very similar to your problem which was stated to be NP-complete by Schaefer in his famous dichotomy theorem paper although he did not give the proof for cubic planar graphs. The problem asks for the existence of two coloring of cubic planar graphs such that every vertex has exactly one neighbor of the same color as itself.