An extremal function for digraph subcontraction

Jagger, Chris

doi:10.1002/(sici)1097-0118(199603)21:3<343::aid-jgt10>3.0.co;2-i

Cited by 8 publications

(7 citation statements)

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“…Recall that a weak minor has the same definition as a strong minor except that we only require that the branch sets induce connected (not necessarily strongly-connected) subgraphs. As mentioned in the Introduction, Jagger [11] investigated average degree conditions for finding weak minors in digraphs. In the context of tournaments, we need the following: Theorem 5.5 (Jagger [11]).…”

Section: Large Out-degree In Tournamentsmentioning

confidence: 99%

“…As mentioned in the Introduction, Jagger [11] investigated average degree conditions for finding weak minors in digraphs. In the context of tournaments, we need the following: Theorem 5.5 (Jagger [11]). There exists an absolute constant C > 0 such that the following holds.…”

Section: Large Out-degree In Tournamentsmentioning

confidence: 99%

“…In particular, he addressed the following question: How many edges must a digraph have in order to guarantee weak (or strong) complete minors as subdigraphs? In particular, he showed [11] that any digraph with average degree Ω(r √ log r) contains a weak − → K r minor. This is a weak-minor analogue of the aforementioned result of Kostochka and Thomason.…”

Section: Introductionmentioning

confidence: 99%

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Strong complete minors in digraphs

Axenovich¹,

Girão²,

Snyder³

et al. 2020

Preprint

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Kostochka and Thomason independently showed that any graph with average degree Ω(r √ log r) contains a Kr minor. In particular, any graph with chromatic number Ω(r √ log r) contains a Kr minor, a partial result towards Hadwiger's famous conjecture. In this paper, we investigate analogues of these results in the directed setting. There are several ways to define a minor in a digraph. One natural way is as follows. A strong − → K r minor is a digraph whose vertex set is partitioned into r parts such that each part induces a strongly-connected subdigraph, and there is at least one edge in each direction between any two distinct parts. We investigate bounds on the dichromatic number and minimum out-degree of a digraph that force the existence of strong − → K r minors as subdigraphs. In particular, we show that any tournament with dichromatic number at least 2r contains a strong − → K r minor, and any tournament with minimum out-degree Ω(r √ log r) also contains a strong − → K r minor. The latter result is tight up to the implied constant, and may be viewed as a strong-minor analogue to the classical result of Kostochka and Thomason. Lastly, we show that there is no function f : N → N such that any digraph with minimum out-degree at least f (r) contains a strong − → K r minor, but such a function exists when considering dichromatic number.

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Section: Large Out-degree In Tournamentsmentioning

confidence: 99%

Section: Large Out-degree In Tournamentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Strong complete minors in digraphs

Axenovich¹,

Girão²,

Snyder³

et al. 2020

Preprint

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“…The containment of these different minors in dense digraphs as well as their relation to the dichromatic number have already been studied in several previous works, see e.g. [2,8,11] for strong minors, [3,7,12,16] for butterfly minors and [1,4,5,13,14,15,20] for topological minors.…”

Section: Introductionmentioning

confidence: 99%

Complete minors in digraphs with given dichromatic number

Mészáros,

Steiner

2021

Preprint

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The dichromatic number χ(D) of a digraph D is the smallest k for which it admits a k-coloring where every color class induces an acyclic subgraph. Inspired by Hadwiger's conjecture for undirected graphs, several groups of authors have recently studied the containment of directed graph minors in digraphs with given dichromatic number. In this short note we improve several of the existing bounds and prove almost linear bounds by reducing the problem to a recent result of Postle on Hadwiger's conjecture.

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“…For a good survey on the relationship between the minor's existence in G and the generalization of the coloring notion to the digraphs we refer the reader to [5], where an oriented version of Hadwiger's conjecture is given, too. Recently, Jagger [6] has shown that if p is large enough, then any digraph on n vertices having at least 10 5 pi/log2 p • n arcs is contractible onto *KP. Nevertheless, this nice asymptotical result does not give a right information about the "little" cases.…”

Section: Introductionmentioning

confidence: 99%