Question 59
Some of the edges of a regular polyhedron are painted red. Such a colouring is called good if, for each vertex of the polyhedron, there is an edge meeting at that vertex that is not painted red. On the other hand, a colouring in which some edges of a regular polyhedron are painted red is called completely good if, in addition to being good, no face of the polyhedron has all its edges painted red. For which regular polyhedra is the maximum number of edges that can be painted in a good colouring equal to the maximum number in a completely good colouring? Justify the answer.