Let G be a graph on n vertices.An induced subgraph H of G is called heavy if there exist two nonadjacent vertices in H with degree sum at least n in G.We say that G is H-heavy if every induced subgraph of G isomorphic to H is heavy.For a family H of graphs, G is called H-heavy if G is H-heavy for every H ∈ H.In this talk we will first introduce a result characterizing all connected graphs R and S other than P3 (the path on three vertices) such that every 2-connected {R, S}-heavy graph is Hamiltonian.This extends several previous results on forbidden subgraph conditions for Hamiltonian graphs.Then we will introduce other three results on characerizing heavy subgraph pairs for the traceability, homogeneous traceability and pancyclicity of graphs.Some further related research problems will also be mentioned.