A note on some embedding problems for oriented graphs

TL;DR

This paper explores conjectures related to embedding problems in oriented graphs, focusing on Hamilton cycles, perfect packings of transitive triangles, and connections to Ramsey numbers, proposing new bounds and conjectures.

Contribution

It introduces new conjectures on minimum semidegree conditions for Hamilton cycle squares and perfect packings, linking these to Ramsey theory in oriented graphs.

Findings

Proposes a conjecture that high semidegree ensures Hamilton cycle squares.

Suggests bounds for perfect packings of transitive triangles.

Explores links between Ramsey numbers and transitive tournament packings.

Abstract

We conjecture that every oriented graph $G$ on $n$ vertices with $\delta ^+ (G) , \delta ^- (G) \geq 5n/12$ contains the square of a Hamilton cycle. We also give a conjectural bound on the minimum semidegree which ensures a perfect packing of transitive triangles in an oriented graph. A link between Ramsey numbers and perfect packings of transitive tournaments is also considered.