The dominated chromatic number of middle graphs

Abstract

A dominated coloring of a graph is a proper vertex coloring such that every color class is dominated with at least one vertex. The minimum number of colors needed for a dominated coloring of a graph G is the dominated chromatic number of G. The middle graph M(G) of a graph G is the graph obtained by subdividing each edge of G exactly once and joining all these newly introduced vertices of adjacent edges of G. For a graph G without isolated vertices, the dominated chromatic number of M(G) is completely determined.