H-linkage of balanced bipartite digraphs under semi-degree condition

This paper studies the H -linked property in balanced bipartite digraphs, generalizing Dirac-type conditions for the Hamiltonian property. We prove that for every ε > 0 there exist constants α 0 , β 0 > 0 such that, for all 0 < α ≤ α 0 and 0 < β ≤ β 0 , there is a constant 𝐶 = 𝐶 ⁡ ( 𝛼 , 𝛽 , 𝜀 ) with the following property. For every balanced bipartite digraph H with k arcs and minimum degree δ ( H ) ≥ 1, every balanced bipartite digraph 𝐷 = 𝐷 ⁡ [ 𝑉 1 , 𝑉 2 ] with | 𝑉 1 ⁢ | = | ⁢ 𝑉 2 ⁢ | = 𝑛 ≥ 𝐶 ⁢ 𝑘 and minimum semi-degree 𝛿 0 ⁡ ( 𝐷 ) ≥ ( 1 / 2 + 𝜀 ) ⁢ 𝑛 is ( α, β )-arbitrary Hamiltonian H -linked. More precisely, for every bipartition-preserving injection f : V ( H ) → V ( D ) and every compatible sequence of prescribed path lengths 𝑛 1 , … , 𝑛 𝑘 with n i ≥ 4 and ∑ 𝑛 𝑖 < 𝛼 ⁢ 𝑛 𝑛 𝑖 ≤ 𝛽 ⁢ 𝑛 , the map f extends to a spanning H -subdivision in which the path replacing the i th arc has exactly n i arcs. The proof uses the absorbing method, combining a connecting lemma for short directed paths, a probabilistic construction of absorber families, and a Hamilton cycle in the remaining digraph. The result extends recent work on arbitrary Hamiltonian H -linked digraphs to the balanced bipartite setting and has consequences for disjoint cycles and spanning subdivisions.