Disconnected forbidden pairs force supereulerian graphs to be hamiltonian

A graph is said to be supereulerian if it has a spanning eulerian subgraph, i.e., a spanning connected even subgraph. A graph is called hamiltonian if it contains a spanning cycle. A graph is said to be $\{R,S\}$-free if it does not contain R or S as an induced subgraph. Yang et al. characterized all pairs of connected graphs $R,S$ such that every supereulerian $\{R,S\}$-free graph is hamiltonian. In this paper, we consider disconnected forbidden graph $R,S$. We characterize all pairs of disconnected graphs $R,S$ such that every supereulerian $\{R,S\}$-free graph of sufficiently large order is hamiltonian. Applying this result, we also characterize all forbidden pairs for the existence of a Hamiltonian cycle in 2-edge connected graphs.