Arbitrary $H$-linked oriented graphs

Abstract: Suppose that $D$ is a digraph, and $H$ is a multi-digraph on $k$ vertices with $q$ arcs. Let $\mathcal{P}(D)$ be the set of paths in a digraph $D$. An $H$-subdivision $(f,g)$ in a digraph $D$ is a pair of bijections $f : V(H)\rightarrow V(D)$ and $g : A(H) \rightarrow \mathcal{P}(D)$ such that for every arc $uv\in A(H)$, $g(uv)$ 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 $n_0 = n_0(k,q)$ such that if $D$ is an oriented graph of order $n\geq n_0$ 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.