Sumner's conjecture

Sumner's conjecture (also called Sumner's universal tournament conjecture) states that every orientation of every $$n$$-vertex tree is a subgraph of every $$(2n-2)$$-vertex tournament. David Sumner, a graph theorist at the University of South Carolina, conjectured in 1971 that tournaments are universal graphs for polytrees. The conjecture was proven for all large $$n$$ by Daniela Kühn, Richard Mycroft, and Deryk Osthus.

Examples
Let polytree $$P$$ be a star $$K_{1,n-1}$$, in which all edges are oriented outward from the central vertex to the leaves. Then, $$P$$ cannot be embedded in the tournament formed from the vertices of a regular $$2n-3$$-gon by directing every edge clockwise around the polygon. For, in this tournament, every vertex has indegree and outdegree equal to $$n-2$$, while the central vertex in $$P$$ has larger outdegree $$n-1$$. Thus, if true, Sumner's conjecture would give the best possible size of a universal graph for polytrees.

However, in every tournament of $$2n-2$$ vertices, the average outdegree is $$n-\frac{3}{2}$$, and the maximum outdegree is an integer greater than or equal to the average. Therefore, there exists a vertex of outdegree $$\left\lceil n-\frac{3}{2}\right\rceil=n-1$$, which can be used as the central vertex for a copy of $$P$$.

Partial results
The following partial results on the conjecture have been proven.
 * There is a function $$f(n)$$ with asymptotic growth rate $$f(n)=2n+o(n)$$ with the property that every $$n$$-vertex polytree can be embedded as a subgraph of every $$f(n)$$-vertex tournament. Additionally and more explicitly, $$f(n)\le 3n-3$$.
 * There is a function $$g(k)$$ such that tournaments on $$n+g(k)$$ vertices are universal for polytrees with $$k$$ leaves.
 * There is a function $$h(n,\Delta)$$ such that every $$n$$-vertex polytree with maximum degree at most $$\Delta$$ forms a subgraph of every tournament with $$h(n,\Delta)$$ vertices. When $$\Delta$$ is a fixed constant, the asymptotic growth rate of $$h(n,\Delta)$$ is $$n+o(n)$$.
 * Every "near-regular" tournament on $$2n-2$$ vertices contains every $$n$$-vertex polytree.
 * Every orientation of an $$n$$-vertex caterpillar tree with diameter at most four can be embedded as a subgraph of every $$(2n-2)$$-vertex tournament.
 * Every $$(2n-2)$$-vertex tournament contains as a subgraph every $$n$$-vertex arborescence.

Related conjectures
conjectured that every orientation of an $$n$$-vertex path graph (with $$n\ge 8$$) can be embedded as a subgraph into every $$n$$-vertex tournament. After partial results by, this was proven by.

Havet and Thomassé in turn conjectured a strengthening of Sumner's conjecture, that every tournament on $$n+k-1$$ vertices contains as a subgraph every polytree with at most $$k$$ leaves. This has been confirmed for almost every tree by Mycroft and.

conjectured that, whenever a graph $$G$$ requires $$2n-2$$ or more colors in a coloring of $$G$$, then every orientation of $$G$$ contains every orientation of an $$n$$-vertex tree. Because complete graphs require a different color for each vertex, Sumner's conjecture would follow immediately from Burr's conjecture. As Burr showed, orientations of graphs whose chromatic number grows quadratically as a function of $$n$$ are universal for polytrees.