Three families of toric rings arising from posets or graphs with small class groups

The main objects of the present paper are (i) Hibi rings (toric arising from order polytopes posets), (ii) stable set perfect graphs), and (iii) edge graphs satisfying odd cycle condition). goal is to analyze those three toric discuss their structures in case where class groups have small rank. We prove that (i), torsionfree. More precisely, we give descriptions groups. Moreover, characterize posets or whose associated rank 1 2. By using characterizations, differences isomorphic classes with