The buckling and postbuckling behavior of thin toroidal shell segments composed of auxetic core and graphene-reinforced face sheets under radial loads is reported in the present research combining exiting analytical solutions with the new material designs. Three types of graphene distribution of laminated face sheets and the lattice auxetic core are considered for convex, concave toroidal shell segments and cylindrical shells. The honeycomb lattice auxetic core can be modeled applying a homogenization technique. The Stein and McElman approximation can be used for longitudinally shallow shells to establish the nonlinear equilibrium equations in the framework of the Donnell shell theory with geometrically nonlinearities taking into account the two-parameter foundation model. The expressions of the radial load-maximal deflection postbuckling curves are achieved using the Galerkin method. The numerical investigations indicate the remarkably positive effects of honeycomb auxetic core and graphene-reinforced face sheets on nonlinear buckling responses of shells.