On ${\mathrm{b}}^{\ast }$-Coloring and $z$-Coloring of graphs with high girth

Zahra Ahmadidahr, Manouchehr Zaker

In a proper vertex coloring $c$ of a graph $G$, a vertex $u$ is called a b-vertex if $u$ is adjacent to a vertex in every other color class. A ${\mathrm{b}}^{\ast }$-coloring is a proper…