Vertex-coloring with defects

Defective coloring is a variant of the traditional vertex-coloring in which adjacent vertices are allowed to have the same color, as long as the induced monochromatic components have a certain structure. Due to its important applications, as for example in the bipartisation of graphs, this type of coloring has been extensively studied, mainly with respect to the size, degree, diameter, and acyclicity of the monochromatic components. We focus on defective colorings with k colors in which the monochromatic components are acyclic and have small diameter, namely we consider (edge, k)-colorings, in which the monochromatic components have diameter 1, and (star, k)-colorings, in which they have diameter 2. We prove that the (edge, 3)-coloring problem remains NP-complete even for graphs with maximum vertex-degree 6, hence answering an open question posed by Cowen et al. [9], and for planar graphs with maximum vertex-degree 7, and we prove that the (star, 3)-coloring problem is NP-complete even for planar graphs with bounded maximum vertex-degree. On the other hand, we give linear-time algorithms for testing the existence of (edge, 2)-colorings and of (star, 2)-colorings of partial 2-trees. Finally, we prove that outerpaths, a notable subclass of outerplanar graphs, always admit (star, 2)-colorings.