Transversal and Hamiltonicity in a bipartite graph collection

TL;DR

This paper establishes minimum degree conditions that guarantee the existence of Hamiltonian paths and Hamiltonian connectivity in bipartite graph collections, improving previous results.

Contribution

It provides new minimum degree criteria for Hamiltonian transversals and connectivity in bipartite graph collections, extending prior work.

Findings

Minimum degree conditions for Hamiltonian transversals

Conditions for Hamiltonian connectivity in bipartite collections

Nearly balanced bipartite graph results

Abstract

Let $\mathbf{G}=\{G_1,\dots,G_{s}\}$ be a collection of $s$ bipartite graphs with the same bipartition $V=(X,Y)$. For a path $P$ with $V(P)=V$ and $|E(P)|=s$, if there exists an injection $\phi$: $E(P)\rightarrow [s]$ such that $e\in E(G_{\phi(e)})$ for each $e\in E(P)$, then we say that the Hamiltonian path $P$ is a $\mathbf{G}$-transversal. 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 that ensure the existence of a $\mathbf{G}$-transversal isomorphic to a Hamiltonian path and the Hamiltonian connectivity of a balanced bipartite graph collection $\mathbf{G}$, which improve the results of [Hu, Li, Li and Xu, Discrete Math., 2024]. Moreover, we also provide a minimum degree condition that guarantees a nearly balanced bipartite graph collection $\mathbf{G}$ contains a $\mathbf{G}$-transversal isomorphic to a Hamiltonian path.