# Subdivisions of oriented cycles in digraphs with large chromatic number

@article{Cohen2016SubdivisionsOO, title={Subdivisions of oriented cycles in digraphs with large chromatic number}, author={Nathann Cohen and Fr{\'e}d{\'e}ric Havet and William Lochet and Nicolas Nisse}, journal={Journal of Graph Theory}, year={2016}, volume={89}, pages={439 - 456} }

An oriented cycle is an orientation of a undirected cycle. We first show that for any oriented cycle C, there are digraphs containing no subdivision of C (as a subdigraph) and arbitrarily large chromatic number. In contrast, we show that for any C a cycle with two blocks, every strongly connected digraph with sufficiently large chromatic number contains a subdivision of C. We prove a similar result for the antidirected cycle on four vertices (in which two vertices have out‐degree 2 and two…

## 16 Citations

### Cycles with two blocks in k‐chromatic digraphs

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- 2018

This paper is able to find a digraph which shows that the answer to the above problem is no and it is shown that if in addition £D is Hamiltonian, then its underlying simple graph is $(k+\ell-1)$-degenerate and thus the chromatic number of $D$ is at most $k-\ell$, which is tight.

### Four Blocks Cycles C(k,1,1,1) in Digraphs

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A four blocks cycle C(k1, k2, k3, k4) is an oriented cycle formed by the union of four internally disjoint directed paths of lengths k1, k2, k3, and k4 respectively. El Mniny[2] proved that if D is a…

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We show that for every η > 0 every suﬃciently large n -vertex oriented graph D of minimum semidegree exceeding (1 + η ) k 2 contains every balanced antidirected tree with k edges and bounded maximum…

### Substructures in digraphs

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The main purpose of the thesis was to exhibit sufficient conditions on digraphs to find subdivisions of complex structures. While this type of question is pretty well understood in the case of…

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

- MathematicsElectron. J. Comb.
- 2019

If $D$ is an oriented path, or an in-arborescence, or the union of two directed paths from x to y and a directed path from y tox, then every digraph with minimum out-degree large enough contains a subdivision of $D$.

### 05C15 Coloring of graphs and hypergraphs 05C10 Planar graphs; geometric and topological aspects of graph theory 05C20 Directed graphs (digraphs), tournaments

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We investigate bounds on the dichromatic number of digraphs which avoid a fixed digraph as a topological minor. For a digraph F , denote by mader− →χ (F ) the smallest integer k such that every…

### On the Complexity of Finding Internally Vertex-Disjoint Long Directed Paths

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The (parameterized) complexity of several problems consisting in deciding whether a given digraph contains a subdivision of a spindle is studied, which generalize both the Maximum Flow and Longest Path problems.

### Bispindle in strongly connected digraphs with large chromatic number

- MathematicsElectron. Notes Discret. Math.
- 2017

### Bispindles in Strongly Connected Digraphs with Large Chromatic Number

- MathematicsElectron. J. Comb.
- 2018

It is conjectured that for any positive integers $k_1, k_2,k_3$, there is an integer g(k-1,k-2, k-3) such that every strongly connected digraph with chromatic number greater than g contains a subdivision of $B(k_2; k_3)$.

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