On b-vertex and b-edge critical graphs

Abstract

A $b$-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the $b$-chromatic number $b(G)$ of a graph $G$ is the largest integer $k$ such that $G$ admits a $b$-coloring with $k$ colors. A simple graph $G$ is called $b^{+}$-vertex (edge) critical if the removal of any vertex (edge) of $G$ increases its b-chromatic number. In this note, we explain some properties in $b^{+}$-vertex (edge) critical graphs, and we conclude with two open problems.