The bb-chromatic number and ff-chromatic vertex number of regular graphs

The bb-chromatic number of a graph GG, denoted by b(G)b(G), is the largest positive integer kk such that there exists a proper coloring for G with kk colors in which every color class contains at least one vertex adjacent to some vertex in each of the other color classes, such a vertex is called a dominant vertex. The ff-chromatic vertex number of a dd-regular graph GG, denoted by f(G)f(G), is the maximum number of dominant vertices of distinct colors in a proper coloring with d+1d+1 colors. El Sahili and Kouider conjectured that b(G)=d+1b(G)=d+1 for any dd-regular graph GG of girth 5. Blidia, Maffray and Zemir (2009) reformulated this conjecture by excluding the Petersen graph and proved it for d≤6d≤6. We study El Sahili and Kouider conjecture by giving some partial answers under supplementary conditions.