Long antipaths and anticycles in oriented graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910293391572992 |
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| author | Chen, Bin Hou, Xinmin Zhou, Hongyu |
| author_facet | Chen, Bin Hou, Xinmin Zhou, Hongyu |
| contents | Let $δ^{0}(D)$ be the minimum semi-degree of an oriented graph $D$. Jackson (1981) proved that every oriented graph $D$ with $δ^{0}(D)\geq k$ contains a directed path of length $2k$ when $|V(D)|>2k+2$, and a directed Hamilton cycle when $|V(D)|\le 2k+2$. Stein~(2020) further conjectured that every oriented graph $D$ with $δ^{0}(D)>k/2$ contains any orientated path of length $k$. Recently, Klimousová and Stein (DM, 2023) introduced the minimum pseudo-semi-degree $\tildeδ^0(D)$ (a slight weaker than the minimum semi-degree condition as $\tildeδ^0(D)\ge δ^0(D))$ and showed that every oriented graph $D$ with $\tildeδ^{0}(D)\ge (3k-2)/4$ contains each antipath of length $k$ for $k\geq 3$. In this paper, we improve the result of Klimousová and Stein by showing that for all $k\geq 2$, every oriented graph with $\tildeδ^0(D)\ge(2k+1)/3$ contains either an antipath of length at least $k+1$ or an anticycle of length at least $k+1$. Furthermore, we answer a problem raised by Klimousová and Stein in the negative. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_05205 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Long antipaths and anticycles in oriented graphs Chen, Bin Hou, Xinmin Zhou, Hongyu Combinatorics 05C20, 05C38 Let $δ^{0}(D)$ be the minimum semi-degree of an oriented graph $D$. Jackson (1981) proved that every oriented graph $D$ with $δ^{0}(D)\geq k$ contains a directed path of length $2k$ when $|V(D)|>2k+2$, and a directed Hamilton cycle when $|V(D)|\le 2k+2$. Stein~(2020) further conjectured that every oriented graph $D$ with $δ^{0}(D)>k/2$ contains any orientated path of length $k$. Recently, Klimousová and Stein (DM, 2023) introduced the minimum pseudo-semi-degree $\tildeδ^0(D)$ (a slight weaker than the minimum semi-degree condition as $\tildeδ^0(D)\ge δ^0(D))$ and showed that every oriented graph $D$ with $\tildeδ^{0}(D)\ge (3k-2)/4$ contains each antipath of length $k$ for $k\geq 3$. In this paper, we improve the result of Klimousová and Stein by showing that for all $k\geq 2$, every oriented graph with $\tildeδ^0(D)\ge(2k+1)/3$ contains either an antipath of length at least $k+1$ or an anticycle of length at least $k+1$. Furthermore, we answer a problem raised by Klimousová and Stein in the negative. |
| title | Long antipaths and anticycles in oriented graphs |
| topic | Combinatorics 05C20, 05C38 |
| url | https://arxiv.org/abs/2401.05205 |