On the $2$-dimensional flow number of the Flower snarks

Abstract

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 $f:E(G)\to {\mathbb{R}}^d$, 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 ${\varphi }_d(G)=\inf \{r:G$ 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 ${\varphi }_2(J_n)\in [1+2\sin \frac{5}{22}\pi ,2.387893647]$, where $J_n$ denotes the Flower snark on $4n$ vertices.