Complete directed minors and chromatic number

Mészáros, Tamás; Steiner, Raphael

doi:10.1002/jgt.22844

Cited by 2 publications

(5 citation statements)

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“…The following is a main result from [16]. By considering cycle hypergraphs of digraphs, we will show that it is an immediate consequence of Theorem 1.…”

Section: Theorem 4 There Exists An Absolute Constant C > 0 Such That ...mentioning

confidence: 78%

“…If D is a digraph, t ≥ 2 an integer, and D does not contain ←→ K t as a strong minor, then D has dichromatic number at most h(t) ≤ 2g(t). [16] whether or not every digraph D with no ←→ K t strong minor has dichromatic number at most t. Inspired by our lower bound on h(t) from Proposition 2, we will actually show that this is not the case: Proposition 2. For every t ≥ 2 there is a digraph D with no strong ←→ K t -minor and dichromatic number at least 3 2 (t − 1) .…”

Section: Theorem 4 There Exists An Absolute Constant C > 0 Such That ...mentioning

confidence: 95%

“…In our third and last application, we give a short reproof of the main result from [16] on coloring digraphs with excluded strong complete minors. The research on this topic was initiated by Axenovich, Girão, Snyder and Weber [3], and then further addressed by Mészáros and the author [16]. To state the result, we adopt the following terminology from [3,16]:…”

Section: Theorem 4 There Exists An Absolute Constant C > 0 Such That ...mentioning

confidence: 99%

“…The research on this topic was initiated by Axenovich, Girão, Snyder and Weber [3], and then further addressed by Mészáros and the author [16]. To state the result, we adopt the following terminology from [3,16]:…”

Section: Theorem 4 There Exists An Absolute Constant C > 0 Such That ...mentioning

confidence: 99%

“…We start by giving the (joint 1 ) proof of Theorem 1 and Proposition 1. The proof idea is inspired by a similar method used in the work [16] by Mészáros and the author to decompose the hypergraph into connected and 2-colorable pieces and derive a graph form that partition.…”

Section: Proofs Of Theorem 1 Proposition 1 and Theoremmentioning

confidence: 99%

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Coloring hypergraphs with excluded minors

Steiner¹

2022

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Hadwiger's conjecture, among the most famous open problems in graph theory, states that every graph that does not contain Kt as a minor is properly (t − 1)-colorable.The purpose of this work is to demonstrate that a natural extension of Hadwiger's problem to hypergraph coloring exists, and to derive some first partial results and applications.Generalizing ordinary graph minors to hypergraphs, we say that a hypergraph H 1 is a minor of a hypergraph H 2 , if a hypergraph isomorphic to H 1 can be obtained from H 2 via a finite sequence of the following operations:• Deleting vertices and hyperedges,• Contracting a hyperedge (i.e., merging the vertices of the hyperedge into a single vertex). First we show that a weak extension of Hadwiger's conjecture to hypergraphs holds true: For every t ≥ 1, there exists a finite (smallest) integer h(t) such that every hypergraph with no Kt-minor is h(t)-colorable, and we prove 3where g(t) denotes the maximum chromatic number of graphs with no Kt-minor. Using the recent result by Delcourt and Postle that g(t) = O(t log log t), this yields h(t) = O(t log log t).We further conjecture that h(t) = 3 2 (t − 1) , i.e., that every hypergraph with no Ktminor is 3 2 (t − 1) -colorable for all t ≥ 1, and prove this conjecture for all hypergraphs with independence number at most 2.By considering special classes of hypergraphs, the above additionally has some interesting applications for ordinary graph coloring, such as:• every graph G is O(kt log log t)-colorable or contains a Kt-minor model all whose branchsets are k-edge-connected, • every graph G is O(qt log log t)-colorable or contains a Kt-minor model all whose branchsets are modulo-q-connected (i.e., every pair of vertices in the same branch-set has a connecting path of prescribed length modulo q), • by considering cycle hypergraphs of digraphs, we obtain known results on strong minors in digraphs with large dichromatic number as special cases. We also construct digraphs with dichromatic number 3 2 (t − 1) not containing the complete digraph on t vertices as a strong minor, thus answering a question by Mészáros and the author in the negative.

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“…The following is a main result from [16]. By considering cycle hypergraphs of digraphs, we will show that it is an immediate consequence of Theorem 1.…”

Section: Theorem 4 There Exists An Absolute Constant C > 0 Such That ...mentioning

confidence: 78%

Section: Theorem 4 There Exists An Absolute Constant C > 0 Such That ...mentioning

confidence: 95%