Given a positive integer $d$, the class $d$-DIR is defined as all those intersection graphs formed from a finite collection of line segments in ${\mathbb R}^2$ having at most $d$ slopes. Since each slope induces an interval graph, it easily follows for every $G$ in $d$-DIR with clique number at most $\omega$ that the chromatic number $\chi(G)$ of $G$ is at most $d\omega$. We show for every even value of $\omega$ how to construct a graph in $d$-DIR that meets this bound exactly. This partially confirms a conjecture of Bhattacharya, Dvo\v{r}\'ak and Noorizadeh. Furthermore, we show that the $\chi$-binding function of $d$-DIR is $\omega \mapsto d\omega$ for $\omega$ even and $\omega \mapsto d(\omega-1)+1$ for $\omega$ odd. This refutes said conjecture of Bhattacharya, Dvo\v{r}\'ak and Noorizadeh.

翻译：给定正整数$d$，类$d$-DIR定义为所有由$\mathbb R^2$中至多具有$d$个斜率的有限线段集合构成的交图。由于每个斜率导出一个区间图，易得对于每个团数至多为$\omega$的$d$-DIR图$G$，其色数$\chi(G)$不超过$d\omega$。我们证明对于每个偶数$\omega$，如何构造一个达到该边界的$d$-DIR图。这部分证实了Bhattacharya、Dvořák和Noorizadeh的一个猜想。进一步，我们证明$d$-DIR的$\chi$有界函数为：当$\omega$为偶数时$\omega \mapsto d\omega$，当$\omega$为奇数时$\omega \mapsto d(\omega-1)+1$。这反驳了Bhattacharya、Dvořák和Noorizadeh的所述猜想。