Transversal and Hamiltonicity in a bipartite graph collection

Abstract: Let $\mathbf{G}={G_1,\dots,G_{2n-1}}$ be a collection of $2n-1$ bipartite graphs on the same bipartition $V=(X,Y)$ with $|X|=|Y|=n$. For a path $P$ with $V(P)=V$ and $|E(P)|=2n-1$, if there exists an injection $φ$: $E(P)\rightarrow [2n-1]$ such that $e\in E(G_{φ(e)})$ for each $e\in E(P)$, then we say that $P$ is a $\mathbf{G}$-transversal isomorphic to a Hamiltonian path. A bipartite graph collection $\mathbf{G}$ is called Hamiltonian connected if for any two vertices $x\in X$ and $y\in Y$, there exists a $\mathbf{G}$-transversal isomorphic to a Hamiltonian path between $x$ and $y$. In this paper, we give the minimum degree conditions to ensure the existence of a $\mathbf{G}$-transversal isomorphic to a Hamiltonian path and the Hamiltonian connectivity of $\mathbf{G}$, which improve the results of [Hu, Li, Li and Xu, Discrete Math., 2024.]