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A $d$-dimensional nowhere-zero $r$-flow on a graph $G$, an $(r,d)$-NZF fromnow on, is a flow where the value on each edge is an element of $\mathbb{R}^d$whose (Euclidean) norm lies in the interval $[1,r-1]$. Such a notion is anatural generalization of the well-known concept of circular nowhere-zero$r$-flow (i.e.\ $d=1$). For every bridgeless graph $G$, the $5$-flow Conjectureclaims that $\phi_1(G)\leq 5$, while a conjecture by Jain suggests that$\phi_d(G)=1$, for all $d \geq 3$. Here, we address the problem of finding apossible upper-bound also for the remaining case $d=2$. We show that, for allbridgeless graphs, $\phi_2(G) \le 1 + \sqrt{5}$ and that the oriented $5$-cycledouble cover Conjecture implies $\phi_2(G)\leq \tau^2$, where $\tau$ is theGolden Ratio. Moreover, we propose a geometric method to describe an$(r,2)$-NZF of a cubic graph in a compact way, and we apply it in someinstances. Our results and some computational evidence suggest that $\tau^2$could be a promising upper bound for the parameter $\phi_2(G)$ for an arbitrarybridgeless graph $G$. We leave that as a relevant open problem which representsan analogous of the $5$-flow Conjecture in the $2$-dimensional case (i.e.complex case).

On $d$-dimensional nowhere-zero $r$-flows on a graph