Oriented Ramsey numbers of some sparse graphs

TL;DR

This paper investigates conditions under which the oriented Ramsey number of certain sparse, cycle-containing oriented graphs equals their order, extending known results and confirming conjectures for specific graph classes.

Contribution

It proves that the oriented Ramsey number equals the graph's order for a class of graphs formed by combining an antidirected cycle with a directed path, advancing understanding of sparse graph Ramsey numbers.

Findings

Proves r(H)=|H| for graphs formed by an antidirected cycle and a directed path.

Confirms conjecture for specific classes of sparse oriented graphs.

Discusses additional cases of oriented graphs with one cycle.

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.