Nordhaus-Gaddum-type relations of three graph coloring parameters

Let G be a simple graph. A coloring of vertices of G is called (i) a 2-proper coloring if vertices at distance 2 receive distinct colors; (ii) an injective coloring if vertices possessing a common neighbor receive distinct colors; (iii) a square coloring if vertices at distance at most 2 receive distinct colors. In this paper, we study inequalities of Nordhaus-Gaddum type for the 2-proper chromatic number, the injective chromatic number, and the square chromatic number.