Theoretical Computer Science ( IF 0.747 ) Pub Date : 2020-02-12 , DOI: 10.1016/j.tcs.2020.02.007 Lars Jaffke, Paloma T. Lima
A b-coloring of a graph G 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 b-Coloring problem asks whether a graph G has a b-coloring with k colors. The b-chromatic number of a graph G, denoted by , is the maximum number k such that G admits a b-coloring with k colors. We consider the complexity of the b-Coloring problem, whenever the value of k is close to one of two upper bounds on : The maximum degree plus one, and the m-degree, denoted by , which is defined as the maximum number i such that G has i vertices of degree at least . We obtain a dichotomy result for all fixed when k is close to one of the two above mentioned upper bounds. Concretely, we show that if , the problem is polynomial-time solvable whenever and, even when , it is -complete whenever . We furthermore consider parameterizations of the b-Coloring problem that involve the maximum degree of the input graph G and give two -algorithms. First, we show that deciding whether a graph G has a b-coloring with colors is parameterized by . Second, we show that b-Coloring is parameterized by , where denotes the number of vertices of degree at least k.