EDITED as PER ONE ANSWER, EDIT IN CAPS and BOLD
This is a follow up on an ill posed question. My question is this.
Suppose you are given a graph $G$ which is 5-critical (that is, it is 5-chromatic but if an edge or vertex is removed, it becomes 4 colorable), and such that it has an edge $xy$ which if removed makes the resulting graph planar. The question pertains to the planar part of the graph that remains after removal of the special edge, $G-xy$. My belief is that all the vertices in G are even, or, all the vertices in $G-xy$ are even except for $x$ and $y$. Candidate planar graphs to describe $G-xy$ are subgraphs of maximal planar graphs which have all vertices even except two, and maximal planar graphs which are uniquely colorable and which have exactly two vertices of degree 3 (in which case, I believe, although I might be wrong, all other vertices are even). I have two questions.
Are there any other candidates to describe $G-xy$,
Is it true, as I believe, that maximal planar graphs which are uniquely colorable and which have exactly two vertices of degree 3 have all other vertices of even degree? As per Misha Lavrov’s answer, WHAT IF IT HAS EXACTLY TWO VERTICES OF DEGREE 3, BOTH OF THE SAME COLOR
I might have other questions, but let’s start with these. Let’s hope I got this right this time around