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 ${\chi }_{b}\left(G\right)$, 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 ${\chi }_{b}\left(G\right)$: The maximum degree $\mathrm{\Delta }\left(G\right)$ plus one, and the m-degree, denoted by $m\left(G\right)$, 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 for all fixed $k\in \mathbb{N}$ when k is close to one of the two above mentioned upper bounds. Concretely, we show that if $k\in \left\{\mathrm{\Delta }\left(G\right)+1-p,m\left(G\right)-p\right\}$, the problem is polynomial-time solvable whenever $p\in \left\{0,1\right\}$ and, even when $k=3$, it is $\mathsf{NP}$-complete whenever $p\ge 2$. We furthermore consider parameterizations of the b-Coloring problem that involve the maximum degree $\mathrm{\Delta }\left(G\right)$ of the input graph G and give two $\mathsf{FPT}$-algorithms. First, we show that deciding whether a graph G has a b-coloring with $m\left(G\right)$ colors is $\mathsf{FPT}$ parameterized by $\mathrm{\Delta }\left(G\right)$. Second, we show that b-Coloring is $\mathsf{FPT}$ parameterized by $\mathrm{\Delta }\left(G\right)+{\ell }_{k}\left(G\right)$, where ${\ell }_{k}\left(G\right)$ denotes the number of vertices of degree at least k.

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