# A Complexity Dichotomy for Critical Values of the b-Chromatic Number of Graphs

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-chromatic number of a graph G, denoted by χ_b(G), is the maximum number k such that G admits a b-coloring with k colors. We consider the complexity of the problem of deciding whether a graph G has a b-coloring with k colors, whenever the value of k is close to one of two upper bounds on χ_b(G): The maximum degree Δ(G) plus one, and the m-degree, denoted by m(G), which is defined as the maximum number i such that G has i vertices of degree at least i-1. We obtain a dichotomy result stating that for fixed k ∈{Δ(G) + 1 - p, m(G) - p}, the problem is polynomial-time solvable whenever p ∈{0, 1} and, even when k = 3, it is NP-complete whenever p > 2. We furthermore give an FPT-algorithm parameterized by k + ℓ, where ℓ denotes the number of vertices of degree at least k, while we observe that the problem is NP-complete whenever k is unbounded and ℓ = 0.

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