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- W4297786068 abstract "We investigate bounds on the dichromatic number of digraphs which avoid a fixed digraph as a topological minor. For a digraph $F$, denote by $text{mader}_{vec{chi}}(F)$ the smallest integer $k$ such that every $k$-dichromatic digraph contains a subdivision of $F$. As our first main result, we prove that if $F$ is an orientation of a cycle then $text{mader}_{vec{chi}}(F)=v(F)$. This settles a conjecture of Aboulker, Cohen, Havet, Lochet, Moura and Thomass'{e}. We also extend this result to the more general class of orientations of cactus graphs, and to bioriented forests. Our second main result is that $text{mader}_{vec{chi}}(F)=4$ for every tournament $F$ of order $4$. This is an extension of the classical result by Dirac that $4$-chromatic graphs contain a $K_4$-subdivision to directed graphs." @default.
- W4297786068 created "2022-10-01" @default.
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- W4297786068 date "2020-08-22" @default.
- W4297786068 modified "2023-09-26" @default.
- W4297786068 title "Dichromatic number and forced subdivisions" @default.
- W4297786068 doi "https://doi.org/10.48550/arxiv.2008.09888" @default.
- W4297786068 hasPublicationYear "2020" @default.
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