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… 
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