Minimum degree conditions for H-linked graphs

For a fixed multigraph H with vertices w₁, ..., w_m, a graph G is H-linked if for every choice of vertices v₁, ..., v_m in G, there exists a subdivision of H in G such that v_i is the branch vertex representing w_i (for all i). This generalizes the notions of k-linked, k-connected, and k-ordered graphs. Given a connected multigraph H with k edges and minimum degree at least two and n ≥ 7.5 k, we determine the least integer d such that every n-vertex simple graph with minimum degree at least d is H-linked. This value D (H, n) appears to equal the least integer d^′ such that every n-vertex graph with minimum degree at least d^′ is b (H)-connected, where b (H) is the maximum number of edges in a bipartite subgraph of H.