# Sumner's conjecture

David Sumner (a graph theorist at the University of South Carolina) conjectured in 1971 that tournaments are universal graphs for polytrees. More precisely, Sumner's conjecture (also called Sumner's universal tournament conjecture) states that every orientation of every ${\displaystyle n}$-vertex tree is a subgraph of every ${\displaystyle (2n-2)}$-vertex tournament.[1] The conjecture remains unproven; Template:Harvtxt call it "one of the most well-known problems on tournaments."

## Examples

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

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

## Partial results

The following partial results on the conjecture are known.

## Related conjectures

Template:Harvtxt conjectured that every orientation of an ${\displaystyle n}$-vertex path graph (with ${\displaystyle n\geq 8}$) can be embedded as a subgraph into every ${\displaystyle n}$-vertex tournament.[7] After partial results by Template:Harvtxt this was proven by Template:Harvtxt.

Havet and Thomassé[9] in turn conjectured a strengthening of Sumner's conjecture, that every tournament on ${\displaystyle n+k-1}$ vertices contains as a subgraph every polytree with at most ${\displaystyle k}$ leaves.

Template:Harvtxt conjectured that, whenever a graph ${\displaystyle G}$ requires ${\displaystyle 2n-2}$ or more colors in a coloring of ${\displaystyle G}$, then every orientation of ${\displaystyle G}$ contains every orientation of an ${\displaystyle n}$-vertex tree. Because complete graphs require a different color for each vertex, Sumner's conjecture would follow immediately from Burr's conjecture.[10] As Burr showed, orientations of graphs whose chromatic number grows quadratically as a function of ${\displaystyle n}$ are universal for polytrees.

## Notes

1. Template:Harvtxt. However the earliest published citations given by Kühn et al. are to Template:Harvtxt and Template:Harvtxt. Template:Harvtxt cites the conjecture as an undated private communication by Sumner.
2. This example is from Template:Harvtxt.
3. Template:Harvtxt and Template:Harvtxt. For earlier weaker bounds on ${\displaystyle f(n)}$, see Template:Harvtxt, Template:Harvtxt, Template:Harvtxt, Template:Harvtxt, and Template:Harvtxt.
4. In Template:Harvtxt, but jointly credited to Thomassé in that paper.
5. This is a corrected version of Burr's conjecture from Template:Harvtxt.

## References

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