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Dichromatic number and forced subdivisions
Journal of Combinatorial Theory Series B ( IF 1.4 ) Pub Date : 2021-11-12 , DOI: 10.1016/j.jctb.2021.10.002
Lior Gishboliner , Raphael Steiner , Tibor Szabó

We investigate bounds on the dichromatic number of digraphs which avoid a fixed digraph as a topological minor. For a digraph F, denote by maderχ(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 maderχ(F)=v(F). This settles a conjecture of Aboulker, Cohen, Havet, Lochet, Moura and Thomassé. 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 maderχ(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 K4-subdivision to directed graphs.



中文翻译:

双色数和强制细分

我们研究了避免将固定有向图作为拓扑次要的有向图二色数的界限。对于有向图F,表示为制造者χ(F)最小整数k,使得每个k -二色二合图包含F的细分。作为我们的第一个主要结果,我们证明如果F是一个循环的方向,那么制造者χ(F)=v(F). 这解决了 Aboulker、Cohen、Havet、Lochet、Moura 和 Thomassé 的猜想。我们还将这个结果扩展到更一般的仙人掌图方向和双向森林。

我们的第二个主要结果是 制造者χ(F)=4对于每个4 阶的锦标赛F。这是狄拉克经典结果的扩展,即 4 色图包含一个4- 细分到有向图。

更新日期:2021-11-13
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