Dichromatic number and forced subdivisions

We investigate bounds on the dichromatic number of digraphs which avoid a fixed digraph as topological minor. For F , denote by mader ? ? ( ) smallest integer k such that every -dichromatic contains subdivision . As our first main result, we prove if is an orientation cycle then = v This settles conjecture Aboulker, Cohen, Havet, Lochet, Moura and Thomassé. also extend this result to more general class orientations cactus graphs, bioriented forests. Our second 4 for tournament order 4. extension classical Dirac 4-chromatic graphs contain K -subdivision directed graphs.