Forbidden pairs for spanning (closed) trails

In Faudree and Gould (1997), the authors determined all pairs of connected graphs {H₁,H₂} such that any connected {H₁,H₂}-free graph has a spanning path (cycle), i.e., hamiltonian path (cycle). In this paper, we consider a similar problem and determine all pairs of forbidden subgraphs guaranteeing the existence of spanning (closed) trails of connected graphs. Our results show that although the forbidden pairs for the existence of spanning trails are the same as the existence of spanning paths, the forbidden pairs for the existence of spanning closed trails (supereulerian) are much different from those for the existence of spanning cycles (hamiltonian).