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On b-vertex and b-edge critical graphs

Eschouf N. I., Blidia M.

Opuscula Mathematica

2015

Vol. 35, no. 2

171--180

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.

On vertex b-critical trees

Blidia M., Eschouf N. I., Maffray F.

Opuscula Mathematica

2013

Vol. 33, no. 1

19-28

A b-coloring is a proper coloring of the vertices of a graph such that each color class has a vertex that has neighbors of all other colors. The b-chromatic number of a graph G is the largest k such that G admits a b-coloring with k colors. A graph G is b-critical if the removal of any vertex of G decreases the b-chromatic number. We prove various properties of b-critical trees. In particular, we characterize b-critical trees.