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The $b$-Chromatic Number and $f$-Chromatic Vertex Number of Regular Graphs

The $b$-chromatic number of a graph $G$, denoted by $b(G)$, is the largest positive integer $k$ such that there exists a proper coloring for G with $k$ 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 $f$-chromatic vertex number of a $d$-regular graph $G$, denoted by $f(G)$, is the maximum number of dominant vertices of distinct colors in a proper coloring with $d+1$ colors. El Sahili and Kouider conjectured that $b(G)=d+1$ for any $d$-regular graph $G$ of girth 5. We study this conjecture by giving some partial answers under supplementary conditions.