Minor-minimal 6-regular graphs in the Klein bottle

Let K6 denote the class of all 6-regular graphs which admit an embedding into the Klein bottle. Using the characterization of graphs in K6 we find minor minimal graphs in K6. As a corollary we show that (i) every 6-regular Klein bottlal graph contains a 6-connected minor, and (ii) no 6-regular graph admits an embedding in both the torus and the Klein bottle.