Transversal cycles and paths in tournaments
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arXiv
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| Format: | Preprint |
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2024
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| author | Chakraborti, Debsoumya Kim, Jaehoon Lee, Hyunwoo Seo, Jaehyeon |
| author_facet | Chakraborti, Debsoumya Kim, Jaehoon Lee, Hyunwoo Seo, Jaehyeon |
| contents | Thomason [$\textit{Trans. Amer. Math. Soc.}$ 296.1 (1986)] proved that every sufficiently large tournament contains Hamilton paths and cycles with all possible orientations, except possibly the consistently oriented Hamilton cycle. This paper establishes $\textit{transversal}$ generalizations of these classical results. For a collection $\mathbf{T}=\{T_1,\dots,T_m\}$ of not-necessarily distinct tournaments on the common vertex set $V$, an $m$-edge directed subgraph $\mathcal{D}$ with the vertices in $V$ is called a transversal if there exists an bijection $φ\colon E(\mathcal{D})\to [m]$ such that $e\in E(T_{φ(e)})$ for all $e\in E(\mathcal{D})$. We prove that for sufficiently large $n$, there exist transversal Hamilton cycles of all possible orientations possibly except the consistently oriented one. We also obtain a similar result for the transversal Hamilton paths of all possible orientations. These results generalize the classical theorem of Thomason, and our approach provides another proof of this theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14300 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Transversal cycles and paths in tournaments Chakraborti, Debsoumya Kim, Jaehoon Lee, Hyunwoo Seo, Jaehyeon Combinatorics Thomason [$\textit{Trans. Amer. Math. Soc.}$ 296.1 (1986)] proved that every sufficiently large tournament contains Hamilton paths and cycles with all possible orientations, except possibly the consistently oriented Hamilton cycle. This paper establishes $\textit{transversal}$ generalizations of these classical results. For a collection $\mathbf{T}=\{T_1,\dots,T_m\}$ of not-necessarily distinct tournaments on the common vertex set $V$, an $m$-edge directed subgraph $\mathcal{D}$ with the vertices in $V$ is called a transversal if there exists an bijection $φ\colon E(\mathcal{D})\to [m]$ such that $e\in E(T_{φ(e)})$ for all $e\in E(\mathcal{D})$. We prove that for sufficiently large $n$, there exist transversal Hamilton cycles of all possible orientations possibly except the consistently oriented one. We also obtain a similar result for the transversal Hamilton paths of all possible orientations. These results generalize the classical theorem of Thomason, and our approach provides another proof of this theorem. |
| title | Transversal cycles and paths in tournaments |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2407.14300 |