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.

Keywords

• \(b\)-coloring
• \(b\)-chromatic number
• critical graphs