Subdivisions in Digraphs of Large Out-Degree or Large Dichromatic Number

Pierre Aboulker (1) , Nathann Cohen (2) , Frédéric Havet (2) , William Lochet (2) , Phablo F S Moura (3) , Stéphan Thomassé (4, 5)

1 DI-ENS - Département d'informatique - ENS Paris
2 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
3 UNICAMP - Universidade Estadual de Campinas = University of Campinas
4 IUF - Institut universitaire de France
5 MC2 - Modèles de calcul, Complexité, Combinatoire

In 1985, Mader conjectured the existence of a function f such that every digraph with minimum out-degree at least f (k) contains a subdivision of the transitive tournament of order k. This conjecture is still completely open, as the existence of f (5) remains unknown. In this paper, we show that if D is an oriented path, or an in-arborescence (i.e., a tree with all edges oriented towards the root) or the union of two directed paths from x to y and a directed path from y to x, then every digraph with minimum out-degree large enough contains a subdivision of D. Additionally, we study Mader's conjecture considering another graph parameter. The dichromatic number of a digraph D is the smallest integer k such that D can be partitioned into k acyclic subdigraphs. We show that any digraph with dichromatic number greater than 4 m (n − 1) contains every digraph with n vertices and m arcs as a subdivision.

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

Pierre Aboulker, Nathann Cohen, Frédéric Havet, William Lochet, Phablo F S Moura, et al.. Subdivisions in Digraphs of Large Out-Degree or Large Dichromatic Number. The Electronic Journal of Combinatorics, 2019, 26, pp.P3.19. ⟨hal-02275082⟩