A -coloring of a graph is a proper coloring of its vertices such that each color class contains a vertex that has at least one neighbor in all the other color classes. The -Coloring problem asks whether a graph has a -coloring with colors.
The -chromatic number of a graph , denoted by , is the maximum number such that admits a -coloring with colors. We consider the complexity of the -Coloring problem, whenever the value of is close to one of two upper bounds on : The maximum degree plus one, and the -degree, denoted by , which is defined as the maximum number such that has vertices of degree at least . We obtain a dichotomy result stating that for fixed , the problem is polynomial-time solvable whenever and, even when , it is NP-complete whenever .
We furthermore consider parameterizations of the -Coloring problem that involve the maximum degree of the input graph and give two FPT-algorithms. First, we show that deciding whether a graph G has a -coloring with colors is FPT parameterized by . Second, we show that -Coloring is FPT parameterized by , where denotes the number of vertices of degree at least .
This is joint work with Paloma T. Lima.