AI 中文总结
本文证明了Gimbert提出的分圆猜想，由此推出有向度直径问题中最大出度d>1、直径k>2的几乎摩尔有向图不存在。
AI 中文摘要
对于n>2且k>1，定义多项式F_{n,k}(x)=Φ_n(1+x+⋯+x^k)，其中Φ_n表示第n个分圆多项式。Gimbert于1999年提出的分圆猜想，根据n和k精确描述F_{n,k}(x)在有理数域ℚ上的不可约性。Conde、Gimbert、González、Miller和Miret在2014年证实，若分圆猜想成立，将意味着几乎摩尔有向图不存在——这是关于有向度直径问题的著名开放问题。本文证明了分圆猜想，进而表明：对于任意d>1且k>2，不存在最大出度为d、直径为k的几乎摩尔有向图。
英文摘要
For $n>2$ and $k>1$, define the polynomial \[F_{n,k}(x) = Φ_n(1 + x + \cdots + x^k),\] where $Φ_n$ denotes the $n$-th cyclotomic polynomial. The \emph{cyclotomic conjecture} proposed by Gimbert (1999) exactly describes the irreducibility of $F_{n,k}(x)$ over $\mathbb{Q}$ in terms of $n$ and $k$. Conde, Gimbert, González, Miller and Miret (2014) established that the cyclotomic conjecture, if true, would imply the non-existence of almost Moore digraphs - a well-known open question concerning the directed degree-diameter problem. In this article, we prove the cyclotomic conjecture and, as a consequence, show that there are no almost Moore digraphs with maximum out-degree $d$ and diameter $k$ for any $d>1$ and $k>2$.
