Arbitrary H-linked oriented graphs

Suppose that D is a digraph, and H is a multi-digraph on k vertices with q arcs. Let P(D) be the set of paths in a digraph D. An H-subdivision $(f,g)$ in a digraph D is a pair of bijections f : V(H)→ V(D) and g : A(H) → P(D) such that for every arc uv∈ A(H), $g(uv)$ is a path from $f(u)$ to $f (v)$, and distinct arcs map into internally vertex disjoint paths in D. Further, D is arbitrary H-linked if any k distinct vertices in D can be extended to an H-subdivision $(f,g)$, and the length of each subdivision path can be specified as a number of at least four. In this paper, we prove that there exists a positive integer n₀ = n₀(k,q) such that if D is an oriented graph of order n≥ n₀ with minimum semi-degree at least $(3n+3k+6q-3)/8$, then D is arbitrary H-linked. This minimum semi-degree is sharp. Also, we refine the bounds on the semi-degree of sufficiently large arbitrary k-linked oriented graphs, sufficiently large arbitrary l-ordered oriented graphs, and sufficiently large oriented graphs with disjoint cycles of prescribed lengths containing prescribed arcs.