二色数与强迫细分
Dichromatic number and forced subdivisions
摘要：
研究了避免固定有向图作为拓扑子式的有向图的二色数的界。对于有向图$F$，用$\text{mader}_{\vecχ}(F)$表示最小整数$k$，使得每个$k$ -二色有向图包含$F$的一个细分。作为我们的第一个主要结果，我们证明了，如果$F$是一个循环的定向，那么$\text{mader}_{\vecχ}(F)=v(F)$。这就解决了Aboulker、Cohen、Havet、Lochet、Pasta和Pastassé的一个猜想。我们还将这一结果扩展到更一般的仙人掌图的方向类，并bireoriented森林。我们的第二个主要结果是，对于每一个$4$阶的竞赛图$F$，有$\text{mader}_{\vecχ}(F)=4$。这是Dirac关于$4$ -色图包含$K_4$ -向有向图的经典结果的推广.
英文摘要：
We investigate bounds on the dichromatic number of digraphs which avoid a fixed digraph as a topological minor. For a digraph $F$, denote by $\text{mader}_{\vecχ}(F)$ 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 $\text{mader}_{\vecχ}(F)=v(F)$. 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 $\text{mader}_{\vecχ}(F)=4$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
