A
d-dimensional nowhere-zero
r-flow on a graph
G, an
(r,d)-NZF from now on, is a flow where the value on each edge is an element of
whose (Euclidean) norm lies in the interval
. Such a notion is a natural generalization of the well-known concept of circular nowhere-zero
r-flow (i.e.\
d=1). For every bridgeless graph
G, the
5-flow Conjecture claims that
... [Show full abstract] , while a conjecture by Jain suggests that , for all . Here, we address the problem of finding a possible upper-bound also for the remaining case d=2. We show that, for all bridgeless graphs, and that the oriented 5-cycle double cover Conjecture implies , where is the Golden 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 some instances. Our results and some computational evidence suggest that could be a promising upper bound for the parameter for an arbitrary bridgeless graph G. We leave that as a relevant open problem which represents an analogous of the 5-flow Conjecture in the 2-dimensional case (i.e. complex case).