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 $\stackrel{\rightharpoonup }{r}\left(H\right)$, is the smallest integer $N$ such that every tournament on $N$ vertices contains a copy of $H$. Rosenfeld (JCT-B, 1974) conjectured that $\stackrel{\rightharpoonup }{r}\left(H\right)=\left|H\right|$ if $H$ is a cycle of sufficiently large order, which was confirmed for $\left|H\right|\ge 9$ by Zein recently, and so does if $H$ is a path. Note that $\stackrel{\rightharpoonup }{r}\left(H\right)=\left|H\right|$ implies any tournament contains $H$ as a spanning subdigraph, it is interesting to ask when $\stackrel{\rightharpoonup }{r}\left(H\right)=\left|H\right|$ 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 $\stackrel{\rightharpoonup }{r}\left(H\right)=\left|H\right|$ 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.