Vertex-bipancyclicity in a bipartite graph collection

Let G={G(1), . . . ,G(2n)} 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 is an element of [4,2], while G is vertex-bipancyclic if any vertex is an element of X boolean OR Y is contained in a partial G-transversal isomorphic to an l-cycle for each even integer l is an element of [4,2]. Bradshaw in [Transversals and bipancyclicity in bipartite graph families, Electron. J. Comb., 2021] showed that for each i is an element of [2], if d(Gi)(x) > n/2 for each x is an element of X and d(Gi)(y) >= n/2 for each y is an element of 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 delta(G(i)) >= +12 for any i is an element of [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. (c) 2024 Elsevier B.V. All rights reserved.