On vertex b-critical trees

Abstract

Abstract. A b-coloring is a proper coloring of the vertices of a graph such that each color class has a vertex that has neighbors of all other colors. The b-chromatic number of a graph G is the largest k such that G admits a b-coloring with k colors. A graph G is b-critical if the removal of any vertex of G decreases the b-chromatic number. We prove various properties of b-critical trees. In particular, we characterize b-critical trees.

A characterization of b-chromatic and partial Grundy numbers by induced subgraphs

“…In contrast with the b-chromatic number, we have the following property about the b-relaxed number. Note that this proposition was already proved for trees [4].…”

“…The concept of b-critical vertices and b-critical edges has been introduced recently and since five years a large number of articles are considering this subject [1,4,5,9,24]. In this section, we illustrate how this notion is strongly connected with the concept of b-t-atom.…”

A characterization of be-critical trees

“…Further, a characterization of all such graphs is given in [1]. On the other hand, the authors of [3] characterized the trees whose b-chromatic number decreases when any vertex is removed. The graphs for which the b-chromatic number increases upon the removal of any edge (or vertex) were explored in [5].…”

On b-vertex and b-edge critical graphs

“…In this context, Ikhlef Eschouf ( [13]) and Blidia et al ([5]) have characterized the class of P 4 -sparse graphs, quasi-line graphs, P 5 -free graphs and d-regular graphs for which b(G − e) < b(G) holds for every edge e in G. They also proved that deciding if a graph is in this class is NP-hard for general graphs ( [5]), even when it is restricted to the subclass of P 5 -free graphs formed by the graphs that are the union of two split graphs. The same authors [4] have recently characterized trees for which b(G − v) < b(G) holds for each vertex v in G ( [4]). The focus of this paper involves studying the graphs in which removing of any vertex (edge) of G increases its b-chromatic number.…”