Oriented Ramsey numbers of some sparse graphs
Junying Lu, Yaojun Chen

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 be an oriented graph without directed cycle. The oriented Ramsey number of , denoted by , is the smallest integer such that every tournament on vertices contains a copy of . Rosenfeld (JCT-B, 1974) conjectured that if is a cycle of sufficiently large order, which was confirmed for by Zein recently, and so does if is a path. Note that implies any tournament contains as a spanning subdigraph, it is interesting to ask when for being a sparse oriented graph. S\'os (1986) conjectured this is true if 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 for being an oriented graph obtained by…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory
