Abstract
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 FPT-algorithms. First, we show that deciding whether a graph G has a b-coloring with m(G) colors is FPT parameterized by Δ(G). Second, we show that b-Coloring is FPT parameterized by Δ(G)+ℓk(G), where ℓk(G) denotes the number of vertices of degree at least k.
Highlights
Given a set of colors, a proper coloring of a graph is an assignment of a color to each of its vertices in such a way that no pair of adjacent vertices receive the same color
We turn to the proof of Theorem 1(ii), that is, we show that b-Coloring remains NP-complete for k = 3 if k = ∆(G) + 1 − p or k = m(G) − p for any p ≥ 2, based on a reduction due to Sampaio [17]
Since b-Chromatic Number and b-Coloring are known to be NP-complete when k = ∆(G) + 1 [15], we make the following observation which is of relevance to us since in Section 5.2, we show that b-Coloring is FPT parameterized by ∆(G) + k(G)
Summary
Given a set of colors, a proper coloring of a graph is an assignment of a color to each of its vertices in such a way that no pair of adjacent vertices receive the same color. The positive results in our dichotomy theorem provide XP-algorithms to decide whether a graph has a b-coloring with a number of colors that either precisely meets or is one below one of two upper bounds on the b-chromatic number, with the parameter being the number of colors in each of the cases. Towards more “flexible” tractability results, we consider parameterized versions of b-Coloring that involve the maximum degree ∆(G) of the input graph G, but ask for the existence of b-colorings with a number of colors that in general is different from ∆(G) + 1 or ∆(G). One of the crucially used facts in the algorithm of the previous theorem is that if we ask whether a graph G has a b-coloring with k = m(G) colors, the number of vertices of degree at least k is at most k We generalize this setting and parameterize b-Coloring by the maximum degree plus the number of vertices of degree at least k. Throughout the text, proofs of statements marked with “♣” are deferred to the full version [13]
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