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

Nicolas Bousquet (1) , William Lochet (2, 3) , Stéphan Thomassé (3)

1 G-SCOP_OC - Optimisation Combinatoire
2 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
3 MC2 - Modèles de calcul, Complexité, Combinatoire

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.

Mots clés: Tournaments; Domination; Transitive digraphs

Domaines: Combinatoire [math.CO]; Informatique [cs]; Mathématique discrète [cs.DM]

Nicolas Bousquet, William Lochet, Stéphan Thomassé. A proof of the Erdős–Sands–Sauer–Woodrow conjecture. Journal of Combinatorial Theory, Series B, 2019, Elsevier Journal of Combinatorial Theory, Series B, 137, pp.316-319. ⟨10.1016/j.jctb.2018.11.005⟩. ⟨hal-02158330v2⟩