Coloring the hypergraph of maximal cliques of a graph with no long path

We consider the problem of coloring the vertices of a graph so that no maximal clique of size at least two is monocolored. We solve the following question: Given a fixed graph F, does there exist an integer f(F) such that the hypergraph of maximal cliques of any F-free graph can be f(F)-colored? We show that the answer is positive if and only if all components of F are paths. In that case we give an estimate of f(F), and the proof contains a polynomial time algorithm to find such a coloring.