Originator(s): David Sumner (University of South Carolina)
Definitions: An oriented tree is a digraph that is an orientation of a tree.
Conjecture: For n > 1, every tournament of order 2n-2 contains every oriented tree of order n.
Background/motivation:
Comments/Partial results: Havet and Thomassé [HT] proved that all the branchings always arise, where a branching is an oriented tree that has paths from a root vertex to all other vertices. For the problem of having all oriented trees, they show that tournaments of order (7n-5)/2 suffice. The proof uses median orders, where a median order of a tournament T is a transitive tournament on V(T) having the most edges in common with T.
References:
[HT] Havet, F; Thomassé, S.
Median orders of tournaments: a tool for the second neighborhood problem and
Sumner's conjecture.
J. Graph Theory 35 (2000), no. 4, 244--256.