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Rapport (Rapport De Recherche) Année : 2016

Subdivisions in digraphs of large out-degree or large dichromatic number *

Résumé

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.
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Dates et versions

hal-01403921 , version 1 (28-11-2016)

Identifiants

  • HAL Id : hal-01403921 , version 1

Citer

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 *. [Research Report] INRIA Sophia Antipolis - I3S. 2016. ⟨hal-01403921⟩
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