coloring – Four colored planar graphs with two vertices which must be colored the same


It is known that in maximal planar graphs with exactly two odd vertices, these two odd vertices must be colored the same in any four coloring. It is also possible to construct planar graphs with exactly two odd vertices which are not maximal, and such that in any four coloring the two odd vertices must be colored the same (using Hajós constructions). My question is whether all planar graphs such that two vertices must be colored the same have all vertices even except for two. And if so, how is this proved?

As an additional question, are there any references where I could look up and look at some maximal planar graphs with exactly two odd vertices?

Thank you.