Clique-Coloring of $$K_{3,3}$$ K 3 , 3 -Minor Free Graphs

A clique-coloring of a given graph G is a coloring of the vertices of G such that no maximal clique of size at least two is monocolored. The clique-chromatic number of G is the least number of colors for which G admits a clique-coloring. It has been proved that every planar graph is 3-clique colorable and every claw-free planar graph, different from an odd cycle, is 2-clique colorable. In this paper, we generalize these results to $$K_{3,3}$$ -minor free ( $$K_{3,3}$$ -subdivision free) graphs.