An approximate version of Sumner's universal tournament conjecture

Details An approximate version of Sumner's universal tournament conjecture An approximate version of Sumner's universal tournament conjecture

2010-10-21

arxiv.org/abs/1010.4429v1

Mathematics

Combinatorics

36 pages

Scientific paper

Sumner's universal tournament conjecture states that any tournament on $2n-2$ vertices contains a copy of any directed tree on $n$ vertices. We prove an asymptotic version of this conjecture, namely that any tournament on $(2+o(1))n$ vertices contains a copy of any directed tree on $n$ vertices. In addition, we prove an asymptotically best possible result for trees of bounded degree, namely that for any fixed $\Delta$, any tournament on $(1+o(1))n$ vertices contains a copy of any directed tree on $n$ vertices with maximum degree at most $\Delta$.

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