The b-chromatic index of graphs

Abstract

a b s t r a c tA b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to at least one vertex in each other color class. The b-chromatic number of G is the maximum integer b(G) for which G has a b-coloring with b(G) colors. This problem was introduced by Irving and Manlove (1999), where they showed that computing b(G) is N P -hard in general and polynomial-time solvable for trees. A natural question that arises is whether the edge version of this problem is also N P -hard or not. Here, we prove that computing the b-chromatic index of a graph G is N P -hard, even if G is either a comparability graph or a C k -free graph, and give partial results on the complexity of the problem restricted to trees, more specifically, we solve the problem for caterpillars graphs. Although solving problems on caterpillar graphs is usually quite simple, this problem revealed itself to be unusually hard. The presented algorithm uses a dynamic programming approach that combines partial solutions which are proved to exist if, and only if, a particular polyhedron is non-empty.

A probabilistic study of generalized solution concepts in satisfiability testing and constraint programming

“…For example, super solutions have been used to formalize the notion of a robust or stable solution of the Boolean satisfiability problem and the constraint satisfaction problem [13,15]. In the literature of combinatorial optimization, solution concepts with a similar flavor have also been studied, such as those used in the definition of the b-chromatic number problem and other minimaximal/maximinimal problems under a given partial order on the solution space of an underlying graph-theoretic problem [16,18], which has drawn continued interest in graph theory [11] and practice [7]. We note that the notion of a Nash equilibrium in game theory and its generalizations can also be regarded as a solution concept that requires a certain type of robustness.…”

New bounds for the b-chromatic number of vertex deleted graphs

“…The b-chromatic number b(G) is the largest integer k such that G admits a b-coloring with k colors. Since this parameter has been introduced by R. W. Irving and D. F. Manlove [6], it aroused the interest of many researchers as we can see in [5], [2] and [3] and, more recently, in [7].…”

Edge-b-Coloring Trees

“…Intuitively, a b-coloring is a proper coloring that cannot be improved by the described heuristic, and the b-chromatic number b(G) measures the worst possible such coloring. Finding b(G) was proved to be N P-hard in general graphs [7], and remains so even when restricted to bipartite graphs [9], chordal graphs [6], or line graphs [4]. In the same article, Irving and Manlove also introduced a simple upper bound for b(G), defined as follows.…”