Let $G$ be a triangulation (maximal planar graph) which is 4-connected and has exactly four odd vertices, (perhaps none adjacent to each other, but at least two non adjacent to each other), and let $x$, $y$ be two of the odd vertices, non adjacent to each other. I wish to prove the following.
Statement $G$ can be 4-colored in such a way that $x$, $y$ are colored with different colors.
Is this statement true or false, and why? How might one try to prove this?
Following Misha Lavrov's suggestion we found counterexamples. Here is a graph with 14 vertices, 4 of them odd, call them $u, v, x, y$, such that $u, v$ must be colored the same, and $x, y$ must also be colored the same in any 4 coloring. In the picture below, $u, v$ are the topmost two yellow vertices, and $x, y$ are the rightmost two green vertices. I still don't know if the statement I was trying to prove will be true if we add the condition that the four odd vertices must be pairwise non adjacent. The graph was generated with plantri by T. Jensen, and the coloring property was checked with igraph by EGME. Please note that although not rendered as a planar graph , the graph is planar.