Sumner's conjecture

Sumner's conjecture (also called Sumner's universal tournament conjecture) states that every orientation of every -vertex tree is a subgraph of every -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 by Daniela Kühn, Richard Mycroft, and Deryk Osthus. Let polytree be a star , in which all edges are oriented outward from the central vertex to the leaves. Then, cannot be embedded in the tournament formed from the vertices of a regular -gon by directing every edge clockwise around the polygon. For, in this tournament, every vertex has indegree and outdegree equal to , while the central vertex in has larger outdegree . Thus, if true, Sumner's conjecture would give the best possible size of a universal graph for polytrees. However, in every tournament of vertices, the average outdegree is , and the maximum outdegree is an integer greater than or equal to the average. Therefore, there exists a vertex of outdegree , which can be used as the central vertex for a copy of . The following partial results on the conjecture have been proven. There is a function with asymptotic growth rate with the property that every -vertex polytree can be embedded as a subgraph of every -vertex tournament. Additionally and more explicitly, . There is a function such that tournaments on vertices are universal for polytrees with leaves. There is a function such that every -vertex polytree with maximum degree at most forms a subgraph of every tournament with vertices. When is a fixed constant, the asymptotic growth rate of is . Every "near-regular" tournament on vertices contains every -vertex polytree. Every orientation of an -vertex caterpillar tree with diameter at most four can be embedded as a subgraph of every -vertex tournament. Every -vertex tournament contains as a subgraph every -vertex arborescence. conjectured that every orientation of an -vertex path graph (with ) can be embedded as a subgraph into every -vertex tournament.