Trees in tournaments

Abstract

Letf(n) be the smallest integer such that every tournament of orderf(n) contains every oriented tree of ordern. Sumner has just conjectures thatf(n)=2n−2, and F. K. Chung has shown thatf(n)≤(1+o(1))nlog2 n. Here we show thatf(n)≤12n andf(n)≤(4+o(1))n.

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References

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    K. B. Reid, andN. C. Wormald: Embedding orientedn-trees in tournaments,Studia Sci. Math. Hungar. 18 (1983), 377–387.

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    A. G. Thomason: Paths and cycles in tournaments,Trans. Amer. Math. Soc. 296 (1986), 167–180.

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Häggkvist, R., Thomason, A. Trees in tournaments. Combinatorica 11, 123–130 (1991). https://doi.org/10.1007/BF01206356

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AMS subject classification (1980)

  • 05 C 20
  • 05 C 35