Graphs with large clique-chromatic numbers

Abstract

The clique-chromatic number of a graph [Formula: see text], [Formula: see text], is the least number of colors on [Formula: see text] without a monocolored maximal clique of size at least two. If [Formula: see text] is triangle-free, [Formula: see text]; we then consider only graphs with a triangle. Unlike the chromatic number, the clique-chromatic number of a graph is not necessary to be at least those of its subgraphs. Thus, for any family of graphs [Formula: see text], the boundedness of [Formula: see text][Formula: see text] has been investigated. Many families of graphs are proved to have a bounded set of clique-chromatic numbers. In literature, only few families of graphs are shown to have an unbounded set of clique-chromatic numbers, for instance, the family of line graphs. This paper gives another family of graphs with such an unbounded set. These graphs are obtained by the well-known Mycielski’s construction with a certain property of the initial graph.