AI 中文总结
本文推导了Flower snark的二维流数的下界，结合已有数值结果确定其流数范围，丰富了图的流数理论研究。
AI 中文摘要
设$r\geq2$为实数，$d$为正整数。图$G$上的$d$维非零$r$-流，或称$(r,d)$-NZF，是指$G$的一个定向与函数$f\colon E(G)\to \mathbb{R}^d$，使得对所有边$e\in E(G)$，$f(e)$的欧几里得范数落在区间$[1,r-1]$内，且对每个顶点$v\in V(G)$，流入$v$的所有流值之和等于流出的流值之和。图$G$的$d$维流数定义为参数$\phi_d(G)=\inf \{r\colon G$具有$(r,d)$-NZF\}$。本文给出了Flower snark的二维流数的一个下界，结合作者此前的数值结果，证明了$\phi_2(J_n) \in [1 + 2 \sin\frac{5}{22}\pi, 2.387893647]$，其中$J_n$表示有$4n$个顶点的Flower snark。
英文摘要
Let $r\ge 2$ be a real number, $d$ a positive integer. A $d$-dimensional nowhere-zero $r$-flow, or $(r,d)$-NZF, on a graph $G$ is an orientation of $G$ together with a function $f\colon E(G)\to \mathbb{R}^d$, such that for all $e\in E(G)$, the Euclidean norm of $f(e)$ lies in the interval $[1,r-1]$, and for every $v\in V(G)$ the sum of all incoming flow values at $v$ equals the sum of all outgoing ones. The $d$-dimensional flow number of $G$ is the parameter $ϕ_d(G)=\inf \{r\colon G$ has an $(r,d)$-NZF$\}$. In this paper we provide a lower bound for the $2$-dimensional flow number of the the Flower snark. In particular, together with a previous numerical result by the authors, we prove that $ϕ_2(J_n) \in [1 + 2 \sin\frac{5}{22}π, 2.387893647]$, where $J_n$ denotes the Flower snark on $4n$ vertices.
