Vertex-bipancyclicity in a bipartite graph collection

Let G={G1,…,G2n} be a bipartite graph collection on the common vertex bipartition (X,Y) with |X|=|Y|=n. We say that G is bipancyclic if there exists a partial G-transversal isomorphic to an ℓ-cycle for each even integer ℓ∈[4,2n], while G is vertex-bipancyclic if any vertex v∈X∪Y is contained in a partial G-transversal isomorphic to an ℓ-cycle for each even integer ℓ∈[4,2n]. Bradshaw in [Transversals and bipancyclicity in bipartite graph families, Electron. J. Comb., 2021] showed that for each i∈[2n], if dGi(x)>n2 for each x∈X and dGi(y)≥n2 for each y∈Y, then G is bipancyclic, which generalizes a classical result of Schmeichel and Mitchem in [Bipartite graphs with cycles of all even lengths, J. Graph Theory, 1982] on the bipancyclicity of bipartite graphs to the setting of graph transversals. Motivated by their work, we study vertex-bipancyclicity in bipartite graph collections and prove that if δ(Gi)≥n+12 for any i∈[2n], then G is vertex-bipancyclic unless n=3 and G consists of 6 identical copies of a 6-cycle. Moreover, we also show the Hamiltonian connectivity of G.