Transversal and Hamiltonicity in a bipartite graph collection
Menghan Ma, Lihua You, Xiaoxue Zhang
TL;DR
This work investigates transversal structures in a bipartite graph collection of size $2n-1$ on a fixed bipartition $(X,Y)$ with $|X|=|Y|=n$, focusing on transversals isomorphic to Hamiltonian paths and on Hamiltonian connectivity. Using the Joos–Kim auxiliary-digraph technique, the authors define an associated digraph from a partial transversal and derive two lemmas that provide sufficient indegree and neighborhood conditions to guarantee a $\mathbf{G}$-transversal isomorphic to a Hamiltonian path between any $x\in X$ and $y\in Y$. They prove two main theorems: with $\delta(G_i)\ge \left\lceil \frac{n}{2}\right\rceil$ the collection either contains a $\mathbf{G}$-transversal isomorphic to a Hamiltonian path or all graphs are the extremal $K_{\frac{n}{2},\frac{n}{2}}\cup K_{\frac{n}{2},\frac{n}{2}}$ (when $n$ is even); with $\delta(G_i)\ge \left\lceil \frac{n+1}{2}\right\rceil$ the collection is Hamiltonian connected or, when $n$ is odd, each $G_i\in\{F,F'\}$. These results strengthen prior work by Hu et al. by closing the gap for $2n-1$ graphs and providing sharp extremal cases.
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.]