Path Extendable Tournaments
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918005428977664 |
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| author | Zhang, Zan-Bo He, Weihua Broersma, Hajo Zhang, Xiaoyan |
| author_facet | Zhang, Zan-Bo He, Weihua Broersma, Hajo Zhang, Xiaoyan |
| contents | A digraph $D$ is called \emph{path extendable} if for every nonhamiltonian (directed) path $P$ in $D$, there exists another path $P^\prime$ with the same initial and terminal vertices as $P$, and $V(P^\prime) = V (P)\cup \{w\}$ for a vertex $w \in V(D)\setminus V(P)$. Hence, path extendability implies paths of continuous lengths between every vertex pair. In earlier works of C. Thomassen and K. Zhang, it was shown that the condition of small $i(T)$ or positive $π_2(T)$ implies paths of continuous lengths between every vertex pair in a tournament $T$, where $i(T)$ is the irregularity of $T$ and $π_2(T)$ denotes for the minimum number of paths of length $2$ from $u$ to $v$ among all vertex pairs $\{u,v\}$. Motivated by these results, we study sufficient conditions in terms of $i(T)$ and $π_2(T)$ that guarantee a tournament $T$ is path extendable. We prove that (1) a tournament $T$ is path extendable if $i(T)< 2π_2(T)-(|T|+8)/6$, and (2) a tournament $T$ is path extendable if $π_2(T) > (7|T|-10)/36$. As an application, we deduce that almost all random tournaments are path extendable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21653 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Path Extendable Tournaments Zhang, Zan-Bo He, Weihua Broersma, Hajo Zhang, Xiaoyan Combinatorics 05C20, 05C38 G.2.2 A digraph $D$ is called \emph{path extendable} if for every nonhamiltonian (directed) path $P$ in $D$, there exists another path $P^\prime$ with the same initial and terminal vertices as $P$, and $V(P^\prime) = V (P)\cup \{w\}$ for a vertex $w \in V(D)\setminus V(P)$. Hence, path extendability implies paths of continuous lengths between every vertex pair. In earlier works of C. Thomassen and K. Zhang, it was shown that the condition of small $i(T)$ or positive $π_2(T)$ implies paths of continuous lengths between every vertex pair in a tournament $T$, where $i(T)$ is the irregularity of $T$ and $π_2(T)$ denotes for the minimum number of paths of length $2$ from $u$ to $v$ among all vertex pairs $\{u,v\}$. Motivated by these results, we study sufficient conditions in terms of $i(T)$ and $π_2(T)$ that guarantee a tournament $T$ is path extendable. We prove that (1) a tournament $T$ is path extendable if $i(T)< 2π_2(T)-(|T|+8)/6$, and (2) a tournament $T$ is path extendable if $π_2(T) > (7|T|-10)/36$. As an application, we deduce that almost all random tournaments are path extendable. |
| title | Path Extendable Tournaments |
| topic | Combinatorics 05C20, 05C38 G.2.2 |
| url | https://arxiv.org/abs/2504.21653 |