Semi‐Degree Condition for Arbitrary H‐Linked Oriented Graphs

Let be a multidigraph on vertices with arcs. An ‐ subdivision in a digraph is a subdigraph obtained by replacing every arc of with a path from to in such that these paths are pairwise internally vertex‐disjoint. A digraph is arbitrary ‐ linked if, for every injection , there exists an ‐subdivision in such that each vertex is mapped to , and the length of every subdivision path can be arbitrarily specified as an integer . An oriented graph is a digraph without two‐cycles. Keevash, Kühn, and Osthus proved that every sufficiently large oriented graph of order with contains a Hamilton cycle (i.e., a ‐subdivision). Subsequently, Kelly, Kühn, and Osthus showed that such oriented graphs are also arbitrary ‐linked, where is a loop. Motivated by these results, we establish a minimum semi‐degree condition for arbitrary ‐linked oriented graphs: there exists such that every oriented graph of order with is arbitrary ‐linked; specifically, if is a loop, this holds under the weaker condition . The result provides an oriented graph analogue of Wang's conjecture on cycle‐factors in graphs and determines the tight semi‐degree bounds for both strongly Hamiltonian‐connected and arbitrary ‐linked oriented graphs.