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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2412.17500 |
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| _version_ | 1866909673657991168 |
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| author | Lu, Junying Chen, Yaojun |
| author_facet | Lu, Junying Chen, Yaojun |
| contents | Let $H$ be an oriented graph without directed cycle. The oriented Ramsey number of $H$, denoted by $\overrightarrow{r}(H)$, is the smallest integer $N$ such that every tournament on $N$ vertices contains a copy of $H$. Rosenfeld (JCT-B, 1974) conjectured that $\overrightarrow{r}(H)=|H|$ if $H$ is a cycle of sufficiently large order, which was confirmed for $|H|\geq 9$ by Zein recently, and so does if $H$ is a path. Note that $\overrightarrow{r}(H)=|H|$ implies any tournament contains $H$ as a spanning subdigraph, it is interesting to ask when $\overrightarrow{r}(H)=|H|$ for $H$ being a sparse oriented graph. Sós (1986) conjectured this is true if $H$ is a directed path plus an additional edge containing the origin of the path as one end, which was confirmed by Petrović (JGT, 1988). In this paper, we show that $\overrightarrow{r}(H)=|H|$ for $H$ being an oriented graph obtained by identifying a vertex of an antidirected cycle with one end of a directed path. Some other oriented Ramsey numbers for oriented graphs with one cycle are also discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17500 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Oriented Ramsey numbers of some sparse graphs Lu, Junying Chen, Yaojun Combinatorics 05C20 Let $H$ be an oriented graph without directed cycle. The oriented Ramsey number of $H$, denoted by $\overrightarrow{r}(H)$, is the smallest integer $N$ such that every tournament on $N$ vertices contains a copy of $H$. Rosenfeld (JCT-B, 1974) conjectured that $\overrightarrow{r}(H)=|H|$ if $H$ is a cycle of sufficiently large order, which was confirmed for $|H|\geq 9$ by Zein recently, and so does if $H$ is a path. Note that $\overrightarrow{r}(H)=|H|$ implies any tournament contains $H$ as a spanning subdigraph, it is interesting to ask when $\overrightarrow{r}(H)=|H|$ for $H$ being a sparse oriented graph. Sós (1986) conjectured this is true if $H$ is a directed path plus an additional edge containing the origin of the path as one end, which was confirmed by Petrović (JGT, 1988). In this paper, we show that $\overrightarrow{r}(H)=|H|$ for $H$ being an oriented graph obtained by identifying a vertex of an antidirected cycle with one end of a directed path. Some other oriented Ramsey numbers for oriented graphs with one cycle are also discussed. |
| title | Oriented Ramsey numbers of some sparse graphs |
| topic | Combinatorics 05C20 |
| url | https://arxiv.org/abs/2412.17500 |