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-Coloring problem asks whether a graph $G$ has a $b$-coloring with $k$ colors. The $b$-chromatic number of a graph $G$, denoted by ${\chi }_b(G)$, 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 ${\chi }_b(G)$: The maximum degree $\mathrm{\Delta }(G)$ 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 $k \in {\Delta(G) + 1 - p, m(G) - p}$, the problem is polynomial-time solvable whenever $p\in \{0,1\}$ and, even when $k=3$, it is NP-complete whenever $p \ge 2$. We furthermore consider parameterizations of the b-Coloring problem that involve the maximum degree $\mathrm{\Delta }(G)$ of the input graph $G$ and give two FPT-algorithms. First, we show that deciding whether a graph $G$ has a $b$-coloring with $m(G)$ colors is FPT parameterized by $\mathrm{\Delta }(G)$. Second, we show that b-Coloring is FPT parameterized by $\mathrm{\Delta }(G)+{\mathrm{\ell }}_k(G)$, where ${\mathrm{\ell }}_k(G)$ denotes the number of vertices of degree at least $k$.