Median orders of tournaments: A tool for the second neighborhood problem and Sumner's conjecture

  • Authors:

  • Frédéric Havet;Stéphan Thomassé

  • Affiliations:

  • Laboratoire LAPCS, UFR de Mathématiques, Université Claude Bernard, 43, Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France;Laboratoire LAPCS, UFR de Mathématiques, Université Claude Bernard, 43, Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2000

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Abstract

We give a short constructive proof of a theorem of Fisher: every tournament contains a vertex whose second outneighborhood is as large as its first outneighborhood. Moreover, we exhibit two such vertices provided that the tournament has no dominated vertex. The proof makes use of median orders. A second application of median orders is that every tournament of order 2n - 2 contains every arborescence of order n  1. This is a particular case of Sumner's conjecture: every tournament of order 2n - 2 contains every oriented tree of order n  1. Using our method, we prove that every tournament of order (7n - 5)-2 contains every oriented tree of order n. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 244–256, 2000