Critical graphs for clique-coloring

We consider the NP-complete problem of coloring the vertices of a graph so that no maximal clique of size at least two is monocolored. A basic property of graph colorations is that a coloration of a graph is also a good coloration of all subgraphs. This property allows to define various notions of ‘critical graphs’. Nothing like these is true for clique-colorations: first of all, a clique-coloration of G is usually not a clique-coloration of the subgraphs. In this note, we propose some alternative definitions of critical-graph for clique-coloration and we give some properties and families of critical-graphs.