2-dimensional unit vector flows

Abstract

We study $2$-dimensional unit vector flows on graphs, that is, nowhere-zero flows that assign to each oriented edge a unit vector in ${\mathbb{R}}^3$. We give a new geometric characterization of ${\mathbb{S}}^2$-flows on cubic graphs. We also prove that the class of cubic graphs admitting an ${\mathbb{S}}^2$-flow is closed under a natural composition operation, which yields further constructions; in particular, blowing up a vertex into a triangle preserves the existence of an ${\mathbb{S}}^2$-flow. Our second contribution is algebraic: we extend the rank-based approach of [SIAM J. Discrete Math., 29 (2015), pp.~2166--2178] from ${\mathbb{S}}^1$-flows to ${\mathbb{S}}^2$-flows. More precisely, we show that if an ${\mathbb{S}}^2$-flow $\ddot{I}†$ satisfies $\mathrm{rank}(S_{\mathbb{Q}}(\ddot{I}†))\le 2$ and $S_{\mathbb{Q}}(\ddot{I}†)$ is odd-coordinate-free, then the graph admits a nowhere-zero $4$-flow.