Forbidden subgraphs for longest cycles to contain vertices with large degrees

Let G be a graph. For a given graph H, we say that G is H-free if G contains no copies of H as an induced subgraph. Suppose that G is 2-connected, has n vertices, and α is a real number with 0≤α≤1. In this paper, we characterize the connected graphs R such that G being R-free implies that every longest cycle of G passes through all vertices with degree at least αn+O(1) in G.