OME 2009 Problem 3

Some edges are painted in red. We say that a coloring of this kind is good, if for each vertex of the polyhedron, there exists an edge which concurs in that vertex and is not painted red. Moreover, we say that a coloring where some of the edges of a regular polyhedron is completely good, if in addition to being good, no face of the polyhedron has all its edges painted red. What regular polyhedrons is equal the maximum number of edges that can be painted in a good color and a completely good? Explain your answer.