Vertex-bipancyclicity in a bipartite graph collection.

Let G = { G 1 , ... , G 2 n } 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 , 2 n ] , 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 , 2 n ]. Bradshaw in [Transversals and bipancyclicity in bipartite graph families, Electron. J. Comb., 2021] showed that for each i ∈ [ 2 n ] , if d G i (x) > n 2 for each x ∈ X and d G i (y) ≥ n 2 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 (G i) ≥ n + 1 2 for any i ∈ [ 2 n ] , 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. [ABSTRACT FROM AUTHOR]