B-Chromatic Number: Beyond NP-Hardness

The b-chromatic number of a graph G, χb(G), is the largest integer k such that G has a k-vertex coloring with the property that each color class has a vertex which is adjacent to at least one vertex in each of the other color classes. In the b-Chromatic Number problem, the objective is to decide whether χb(G) ≥ k. Testing whether χb(G) = ∆(G) + 1, where ∆(G) is the maximum degree of a graph, itself is NP-complete even for connected bipartite graphs (Kratochvíl, Tuza and Voigt, WG 2002). In this paper we study b-Chromatic Number in the realm of parameterized complexity and exact exponential time algorithms. We show that b-Chromatic Number is W[1]-hard when parameterized by k, resolving the open question posed by Havet and Sampaio (Algorithmica 2013). When k = ∆(G) + 1, we design an algorithm for b-Chromatic Number running in time 2O(k log k)nO(1). Finally, we show that b-Chromatic Number for an n-vertex graph can be solved in time O(3nn4 logn). 1998 ACM Subject Classification F.2 Analysis of Algorithms and Problem Complexity