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Median orders of tournaments: A tool for the second neighborhood problem and Sumner’s conjecture. (English) Zbl 0969.05029
A median order of a tournament \(T=(V,E)\) is an acyclic tournament \(L=(V,A)\) which is a total order of the vertices of \(T\) and which maximizes the number of arcs belonging to \(E\cap A.\) The authors apply median orders of tournaments as a tool to prove two results: (1) a theorem of D. C. Fisher [J. Graph Theory 23, No. 1, 43-48 (1996; Zbl 0857.05042)]: every tournament contains a vertex whose second outneighborhood is as large as its first outneighborhood; and (2) every tournament of order \(\frac{7n-5}{2}\) contains every oriented tree of order \(n\) (a particular case of Sumner’s conjecture; see N. C. Wormald [Lect. Notes Math. 1036, 417-419 (1983; Zbl 0521.05032)]).

MSC:
05C20 Directed graphs (digraphs), tournaments
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