Discussion on the (partial) Grundy and b-chromatic numbers of graphs
Résumé
The b-chromatic and Grundy colorings are two proper colorings of the vertices of a graph which maximize the number of colors used. While the b-chromatic coloring determines, for every color, a vertex adjacent to all the other colors, the Grundy coloring allows each vertex with color c to be adjacent to colors 1 to c-1. The constraints of each coloring imply that they are difficult to compare. The partial Grundy coloring, which have for every color a vertex adjacent to all the lower colors than its, is defined as a relaxed Grundy coloring, and could be see also as a relaxed b-chromatic coloring. From this remark, could it be considered as a point of reference to relate the b-chromatic and Grundy colorings? In this paper, we present three graph parameters, linked to these colorings, and we discuss the relations between them.