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A Complexity Dichotomy for Critical Values of the b-Chromatic Number of Graphs
Theoretical Computer Science ( IF 0.718 ) 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 χb(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 χb(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 for all fixed k∈N when k is close to one of the two above mentioned upper bounds. Concretely, we show that if 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 consider parameterizations of the b-Coloring problem that involve the maximum degree Δ(G) of the input graph G and give two FTP-algorithms. First, we show that deciding whether a graph G has a b-coloring with m(G) colors is FTP parameterized by Δ(G). Second, we show that b-Coloring is FTP parameterized by Δ(G)+ℓk(G), where ℓk(G) denotes the number of vertices of degree at least k.
更新日期:2020-02-12

 

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