On minimum degree implying that a graph is H-linked

Given a fixed multigraph H, possibly containing loops, with V(H) = {h ₁,..., h _m}, we say that a graph G is H-linked if for every choice of m vertices v ₁,...,v _m in G, there exists a subdivision of H in G such that V _i is the branch vertex representing h _i (for all i). This generalizes the concept of k-linked graphs (as well as a number of other well-known path or cycle properties). In this paper we determine a sharp lower bound on δ(G) (which depends upon H) such that each graph G on at least 10(|V(H)| + |E(H)|) vertices satisfying this bound is H-linked.