Transversal and Hamiltonicity in a bipartite graph collection
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| Format: | Preprint |
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2026
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| _version_ | 1866911500200837120 |
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| author | Ma, Menghan You, Lihua Zhang, Xiaoxue |
| author_facet | Ma, Menghan You, Lihua Zhang, Xiaoxue |
| contents | 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 $ϕ$: $E(P)\rightarrow [s]$ such that $e\in E(G_{ϕ(e)})$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_17758 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Transversal and Hamiltonicity in a bipartite graph collection Ma, Menghan You, Lihua Zhang, Xiaoxue Combinatorics 05C38 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 $ϕ$: $E(P)\rightarrow [s]$ such that $e\in E(G_{ϕ(e)})$ 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. |
| title | Transversal and Hamiltonicity in a bipartite graph collection |
| topic | Combinatorics 05C38 |
| url | https://arxiv.org/abs/2601.17758 |