On the $sd_{b}$-critical graphs

A $b$-coloring of a graph\ $G$ is a proper coloring of its vertices such that each color class contains a vertex that has a neighbor in every other color classes. The $b$-chromatic number of a graph $G$, denoted by $b(G)$, is the largest integer $k$ such that $G$ admits a $b$-coloring with $k$ colors. Let $G_{e}$ be the graph obtained from $G$ by subdividing the edge $e $. A graph $G$ is $sd_{b}$-critical if $b(G_{e})<b(G)$ holds for any edge $e$ of $G$. In this paper, we first present \ several basic properties of $sd_{b} $-critical graphs and then we give a characterization of $sd_{b}$-critical $P_{4}$-sparse graphs and $sd_{b}$-critical quasi-line graphs.