2-extendability and kk-resonance of non-bipartite Klein-bottle polyhexes

C. Thomassen classified Klein-bottle polyhexes into five classes [C. Thomassen, Tilings of the torus and the Klein bottle and vertex-transitive graphs on a fixed surface, Trans. Amer. Math. Soc. 323 (1991) 605–635]. In this paper, by implementing cutting and gluing operations, we reclassify the five classes of Klein-bottle polyhexes into two classes—the bipartite case K(p,q,t)K(p,q,t) and the non-bipartite case N(p,q,t)N(p,q,t). Further, we completely characterize 2-extendable and kk-resonant non-bipartite Klein-bottle polyhexes, respectively.