An irregular coloring of a graph is a proper vertex coloring that distinguishes vertices in the graph either by their own colors or by the colors of their neighbors. In algebraic graph theory, graphs with a certain amount of symmetry can sometimes be specified in terms of a group and a smaller graph called a voltage graph. Radcliffe and Zhang found a bound for the irregular chromatic number of a graph on n vertices. In this paper we use voltage graphs to construct graphs achieving that bound.
Anderson, Mark; Vitray, Richard P.; and Yellen, Jay, "Irregular Colorings of Regular Graphs" (2012). Faculty Publications. 86.