Degree Conditions for H -Linked Digraphs

Source abstract

Given a (multi)digraph H , a digraph D is H - linked if every injective function ι: V(H) → V(D) can be extended to an H -subdivision. In this paper, we give sharp degree conditions that ensure a sufficiently large digraph D is H -linked for arbitrary H . The notion of an H -linked digraph extends the classes of m -linked, m -ordered and strongly m -connected digraphs. First, we give sharp minimum semi-degree conditions for H -linkedness, extending results of Kühn and Osthus on m -linked and m -ordered digraphs. It is known that the minimum degree threshold for an undirected graph to be H -linked depends on a partition of the (undirected) graph H into three parts. Here, we show that the corresponding semi-degree threshold for H -linked digraphs depends on a partition of H into as many as nine parts. We also determine sharp Ore–Woodall-type degree-sum conditions ensuring that a digraph D is H -linked for general H . As a corollary, we obtain (previously undetermined) sharp degree-sum conditions for m -linked and m -ordered digraphs.