By the construction of suitable graphs and the determination of their chromatic polynomials, we resolve two open questions concerning real chromatic roots. First we exhibit graphs for which the Beraha number $B_{10}=\left(5+\sqrt{5}\right)∕2$ is a chromatic root. As it was previously known that no other noninteger Beraha number is a chromatic root, this completes the determination of precisely which Beraha numbers can be chromatic roots. Next we construct an infinite family of 3‐connected graphs such that for any $k\ge 1$, there is a member of the family with $q=2$ as a chromatic root of multiplicity at least $k$. The former resolves a question of Salas and Sokal and the latter a question of Dong and Koh.

中文翻译：

---

色根在2和Beraha数B10

通过构造合适的图并确定它们的色多项式，我们解决了有关实色根的两个开放问题。首先，我们展示其贝拉哈数的图${乙}_{10}=（5+\sqrt{5}）∕2$是色根。如先前所知，没有其他非整数贝拉哈数不是色根，这就完成了对哪些贝拉哈数可以是色根的精确确定。接下来，我们构造一个无限的三连通图族，这样对于任何$ķ\ge \mathrm{1个}$，有一个家庭成员 $q=2$ 作为复数的色根 $ķ$。前者解决了萨拉斯和索卡尔的问题，而后者解决了董和Koh的问题。