Small snarks and 6-chromatic triangulations on the Klein bottle

It is known that for every nonorientable surface there are infinitely many (large) snarks that can be polyhedrally embedded on that surface. We take a dual approach to the embedding of snarks on the Klein bottle, and investigate edge-colorings of 6-chromatic triangulations of the Klein bottle. In the process, we discover the smallest snarks that can be polyhedrally embedded on the Klein bottle. Additionally, we show that every triangulation containing certain 6-critical graphs on the Klein bottle must have a Grünbaum coloring and thus cannot admit a dual embedded snark.