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

Abstract : 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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https://hal.archives-ouvertes.fr/hal-02275082
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Submitted on : Thursday, November 7, 2019 - 6:32:30 PM
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  • HAL Id : hal-02275082, version 1

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

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