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Subdivisions of oriented cycles in digraphs with large chromatic number. (English) Zbl 06997246
Summary: 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 vertices have in-degree 2).

05C20 Directed graphs (digraphs), tournaments
05C38 Paths and cycles
05C15 Coloring of graphs and hypergraphs
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