b-coloring of m-tight graphs

A given k-coloring c of a graph G = ( V , E ) is a b-coloring if for every color class c i , 1 ⩽ i ⩽ k , there is a vertex colored i whose neighborhood intersect every other color class c j of c. The b-chromatic number of G is the greatest integer k such that G admits a b-coloring with k colors. A graph G is m-tight if it has exactly m = m ( G ) vertices of degree exactly m − 1 , where m ( G ) is the largest integer m such that G has at least m vertices of degree at least m − 1 . Determining the b-chromatic number of a m-tight graph G is NP-hard [Jan Kratochvil, Zsolt Tuza, and Margit Voigt. On the b-chromatic number of graphs. In Proc. of the 28th International Workshop on Graph-Theoretic Concepts in Computer Science, pages 310–320. Springer-Verlag, 2002]. In this paper, we define the b-closure and the partial b-closure of a m-tight graph. These concepts were used to give a characterization of m-tight graphs whose b-chromatic number is equal to m. To illustrate an application of our characterization, we provide some examples for which we can answer in polynomial time whether χ b ( G ) < m . We also generalized the definition of pivoted tree introduced by Irving and Manlove [Robert W. Irving and David F. Manlove. The b-chromatic number of a graph. Discrete Appl. Math., 91(1-3):127–141, 1999] and show how the existence of our pivots affect the behavior of the b-chromatic number parameter in m-tight graphs.