Characterize forbidden pairs forcing a 2-factorable graph to be hamiltonian

A graph is said to be { R , S } -free if G does not contain R or S as an induced subgraph. In 2020, Yang and Xiong characterized all connected pairs R , S such that every 2-connected { R , S } -free graph who has a 2-factor is hamiltonian. In this paper, we characterize all forbidden pairs (not necessary connected) R , S such that every 2-connected { R , S } -free graph who has a 2-factor is hamiltonian. Besides, we have proved a rather interesting result ( Theorem 11 ): as long as { k K 1 ∪ P 3 , Z 1 } ( k ≥ 3 ) is forbidden, every graph (not necessary 2-connected) admitting a 2-factor of order at least R ( 3 , 3 k − 3 ) is hamiltonian.