Forbidden Pairs of disconnected graphs for traceability and hamiltonicity

In this paper, we firstly characterize all forbidden pairs {R,S} for graphs with a spanning trail that are traceable and traceable graphs that are hamiltonian. There is no change of forbidden pairs for hamiltonicity if we impose a necessary condition of assumption that the graph is traceable; however, there is some difference of forbidden pairs for traceability if we impose a necessary condition that the graph has a spanning trail: different on two pairs of forbidden subgraphs {K_1,3,Z₂∪K₁},{K_1,4,Z₁} (where Z_i is the graph obtained by identifying a vertex of a K₃ with an end-vertex of a P_i+1). As a byproduct, we prove that if G is a connected {K_1,4,Z₁}-free graph, then every subgraph G[T] induced by a trail T is traceable and every subgraph G[T] induced by a closed trail T is either hamiltonian or K₂∨3K₁.