2-extendability of toroidal polyhexes and Klein-bottle polyhexes

A toroidal polyhex (resp. Klein-bottle polyhex) described by a string (p,q,t)(p,q,t) arises from a p×qp×q-parallelogram of a hexagonal lattice by a usual torus (resp. Klein bottle) boundary identification with a torsion tt. A connected graph GG admitting a perfect matching is kk-extendable if |V(G)|≥2k+2|V(G)|≥2k+2 and any kk independent edges can be extended to a perfect matching of GG. In this paper, we characterize 2-extendable toroidal polyhexes and 2-extendable Klein-bottle polyhexes.