The Grötzsch Theorem for the hypergraph of maximal cliques

Abstract

In this paper, we extend the Grötzsch Theorem by proving that the clique hypergraph ℋ(G) of every planar graph is 3-colorable. We also extend this result to list colorings by proving that ℋ(G) is 4-choosable for every planar or projective planar graph G. Finally, 4-choosability of ℋ(G) is established for the class of locally planar graphs on arbitrary surfaces.