Oriented Ramsey numbers of some sparse graphs

Abstract

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. S\'os (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 Petrovi\'{c} (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.