2024/12/23 by Lu, Junying, Chen, Yaojun
#05C20 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2412.17500
Let H be an oriented graph without directed cycle. The oriented Ramsey number of H, denoted by \overrightarrowr(H), is the smallest integer N such that every tournament on N vertices contains a copy of H. Rosenfeld (JCT-B, 1974) conjectured that \overrightarrowr(H)=|H| if H is a cycle of sufficiently large order, which was confirmed for |H|≥ 9 by Zein recently, and so does if H is a path. Note that \overrightarrowr(H)=|H| implies any tournament contains H as a spanning subdigraph, it is interesting to ask when \overrightarrowr(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 \overrightarrowr(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.