Coloring clique-hypergraphs of graphs with no subdivision of K5 K 5
A clique-coloring of a graph G is a coloring of the vertices of G  so that no maximal clique of size at least two is monochromatic. The clique-hypergraph, H(G) H ( G ) , of a graph G  has V(G) V ( G ) as its set of vertices and the maximal cliques of G  as its hyperedges. A (vertex) coloring of H(G) H ( G ) with no monochromatic hyperedge is a clique-coloring of G. The clique-chromatic number of G is the least number of colors for which G  admits a clique-coloring. Every planar graph has been proved to be 3-clique-colorable and every claw-free planar graph, different from an odd cycle, has been proved to be 2-clique-colorable. In this paper we first generalize the result of planar graphs to K5 K 5 -minor-free graphs. Furthermore, we generalize the result of claw-free planar graphs to K5 K 5 -subdivision-free graphs and give a polynomial-time algorithm to find a 2-clique-coloring of K5 K 5 -subdivision-free graphs.
clique-hypergraph clique-coloring planar graph polynomial-time algorithm k5 k 5 -subdivision-free graph k5 k 5 k5 k 5 k 5