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A proof of the Erdős–Sands–Sauer–Woodrow conjecture

Nicolas Bousquet 1 William Lochet 2 Stéphan Thomassé 3, 4
1 G-SCOP_OC - Optimisation Combinatoire
G-SCOP - Laboratoire des sciences pour la conception, l'optimisation et la production
2 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
3 MC2 - Modèles de calcul, Complexité, Combinatoire
LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : A very nice result of Bárány and Lehel asserts that every finite subset X or can be covered by X-boxes (i.e. each box has two antipodal points in X). As shown by Gyárfás and Pálvőlgyi this result would follow from the following conjecture: If a tournament admits a partition of its arc set into k quasi-orders, then its domination number is bounded in terms of k. This question is in turn implied by the Erdős–Sands–Sauer–Woodrow conjecture: If the arcs of a tournament T are coloured with k colour's, there is a set X of at most vertices such that for every vertex v of T, there is a monochromatic path from X to v. We give a short proof of this statement. We moreover show that the general Sands–Sauer–Woodrow conjecture (which as a special case implies the stable marriage theorem) is valid for directed graphs with bounded stability number. This conjecture remains however open.
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Submitted on : Thursday, November 21, 2019 - 10:05:57 AM
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Nicolas Bousquet, William Lochet, Stéphan Thomassé. A proof of the Erdős–Sands–Sauer–Woodrow conjecture. Journal of Combinatorial Theory, Series B, Elsevier, 2019, Elsevier Journal of Combinatorial Theory, Series B, 137, pp.316-319. ⟨10.1016/j.jctb.2018.11.005⟩. ⟨hal-02158330⟩



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