An extremal problem for H-linked graphs

We introduce the notion of H-linked graphs, where H is a fixed multigraph 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 k and n ≥ 5k + 6, we determine the least integer d such that, for every loopless graph H with k edges and minimum degree at least two, every n-vertex graph with minimum degree at least d is H-linked. This value D₁(k, n) appears to equal the least integer d′ such that every n-vertex graph with minimum degree at least d′ is k-connected. On the way to the proof, we extend a theorem by Kierstead et al. on the least integer d″ such that every n-vertex graph with minimum degree at least d″ is k-ordered.