Forbidden pairs for 2-factorable and hamiltonian graphs under the necessary condition

In [1], Wang and Xiong characterize all forbidden pairs (not necessary connected) R,S such that 2-connected {R,S}-free graph G admitting a 2-factor is hamiltonian. To be more comprehensive, in this paper, we characterize all forbidden pairs (not necessary connected) R,S such that connected (or 2-edge-connected) {R,S}-free graph G admitting a 2-factor is hamiltonian. Besides, we characterize all forbidden pairs (not necessary connected) R,S such that connected (or 2-edge-connected) {R,S}-free graph G admitting an even-factor has a 2-factor. Comparing with the main result of Yang and Xiong (2023), we give all disconnected forbidden pairs. In the end, we find all forbidden pairs R,S such that connected (or 2-edge-connected) {R,S}-free graph G who has an even-factor is hamiltonian.