Ore-type degree conditions for a graph to be H-linked

Given a fixed multigraph H with V(H) = {h₁,..., 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 for every i, v_i, is the branch vertex representing h _i. This generalizes the notion of k-linked graphs (as well as some other notions). For a family H of graphs, a graph G is H-linked if G is H-linked for every H ∈ H. In this article, we estimate the minimum integer r=r(n,k,d) such that each n-vertex graph with σ₂(G)≥r is H-linked, where H is the family of simple graphs with k edges and minimum degree at least d≥2.