How to come up with an example of a triangulation with two adjacent vertices of odd degree?

Question

We define a triangulation as a planar graph with all faces(including the outer one) as triangles. What is an example of a triangulation with exactly two vertices of odd degree that are adjacent to each other? The number of vertices of even degree is unimportant.

Answer 1

Vertices: A B C D E F G H I J K L

Edges: AB AC AD AE AL BE BF BG BL CD CH CL DE DH EF EH EI EJ EK FG FK GK GL HI HL IJ IL JK JL KL

The vertices A B H K are of odd degree, but A and B are the only vertices of odd degree that are adjacent.

If you allow only two vertices of odd degree in the whole graph, there is no such graph where they are adjacent.

Proof:

Let's assume the vertices A B C D are part of the smallest possible graph where A and B are the only vertices of odd degree. We remove vertex A and the edges AB AC, then we triangulate the graph by replacing the edge AD by the remaining even number of edges. So A disappeares, B turns even, C and D turn odd and all other verices' grades remain the same. So we have again exactly two adjacent vertices C and D in a smaller graph, which contradicts our assumption that our original graph were the smallest one.