2-dimensional unit vector flows

TL;DR

This paper explores 2D unit vector flows on graphs, providing geometric and algebraic characterizations of flows on cubic graphs, and extends rank-based methods to establish conditions for the existence of nowhere-zero 4-flows.

Contribution

It introduces a new geometric characterization of S^2-flows on cubic graphs and extends algebraic rank-based techniques from S^1 to S^2 flows, including preservation properties.

Findings

A geometric characterization of S^2-flows on cubic graphs.

Closure of cubic graphs with S^2-flows under vertex blow-up operations.

Conditions under which S^2-flows imply the existence of nowhere-zero 4-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 $\varphi$ satisfies $\operatorname{rank}(S_{\mathbb{Q}}(\varphi))\le 2$ and $S_{\mathbb{Q}}(\varphi)$ is odd-coordinate-free, then the graph admits a nowhere-zero $4$-flow.