A cycle C of a graph G is nice if \(G-V(C)\) has a perfect matching. A graph G is called even (odd) cycle-nice if each even (odd) cycle of G is nice. (Wang et al. Discrete Mathematics, 345(7):112876, 2022), (S. Zhang et al. Discrete Mathematics & Theoretical Computer Science 2020) studied 2-connected even cycle-nice claw-free graphs. In this paper, we investigate odd cycle-nice graphs and then only consider non-bipartite graphs. We show that a 2-connected odd cycle-nice non-bipartite graph is factor-critical. We completely characterize the structure of 2-connected odd cycle-nice claw-free non-bipartite graphs by three types of operations.
