Unit vector flows

Definitions and mathematical statement

Let $G$ be a finite undirected bridgeless graph, with parallel edges allowed. Orient its edges arbitrarily. A unit vector flow in $\mathbb R^3$ assigns to every edge a vector $f(e)$ of Euclidean norm one and satisfies \[ \sum_{e\text{ leaving }v}f(e) -\sum_{e\text{ entering }v}f(e)=0 \qquad(v\in V(G)). \] Changing an orientation and negating the associated vector preserves the definition. The first conjecture is that every bridgeless finite graph admits such a flow. The dimension is fixed at three, independent of the graph, and every edge has exactly unit norm.

The source also proposes a spherical formulation. Put $S^2=\{x\in\mathbb R^3:\|x\|=1\}$. Does there exist a function \[ q:S^2\longrightarrow\{-4,-3,-2,-1,1,2,3,4\} \] such that $q(-x)=-q(x)$ for every $x$, and $q(x)+q(y)+q(z)=0$ whenever $x,y,z\in S^2$ satisfy $x+y+z=0$? Such a triple consists of three equally spaced points on a great circle, with pairwise angles $120$ degrees. No continuity, measurability, or other regularity assumption is imposed on $q$.

These statements involve exact conservation, not approximate balancing. In the second one, the condition is imposed simultaneously on all qualifying triples of sphere points, rather than on a finite configuration chosen for one graph. The listed integer values deliberately exclude zero and identify antipodal values by opposite signs.

Short English statement

Do all bridgeless graphs have three-dimensional unit vector flows, and can all sphere directions be assigned small nonzero integers compatibly with antipodes and balanced triples?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3222 (OPG-136), OpenGarden. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Unit vector flows. Original problem and discussion. http://www.openproblemgarden.org/op/unit_vector_flows.