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2-dimensional unit vector flows

Hussein Houdrouge, Bobby Miraftab, Pat Morin

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.

2-dimensional unit vector flows

Abstract

We study -dimensional unit vector flows on graphs, that is, nowhere-zero flows that assign to each oriented edge a unit vector in . We give a new geometric characterization of -flows on cubic graphs. We also prove that the class of cubic graphs admitting an -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 -flow. Our second contribution is algebraic: we extend the rank-based approach of [SIAM J. Discrete Math., 29 (2015), pp.~2166--2178] from -flows to -flows. More precisely, we show that if an -flow satisfies and is odd-coordinate-free, then the graph admits a nowhere-zero -flow.
Paper Structure (10 sections, 27 theorems, 26 equations, 2 figures)

This paper contains 10 sections, 27 theorems, 26 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Geometric Approach of S2-Flows
  4. Examples of Graphs with an S2-Flows
  5. Generalised Peterson Graph
  6. $\mathbb{S}^2$-immersion to One and Two Points
  7. Combining Cubic Graphs with $\mathbb{S}^2$-flows
  8. An Algebraic Approach of S2-Flows
  9. S-n-Flows for n>=3
  10. Further Research

Key Result

Theorem 1

A cubic graph $G$ admits an equiangular $\mathbb{S}^2$-immersion if and only if $G$ admits an $\mathbb{S}^2$-flow.

Figures (2)

  • Figure 1: Splitting a vertex of odd degree.
  • Figure 2: Two different equiangular $\mathbb{S}^2$-immersion of the Petersen graph.

Theorems & Definitions (48)

  • Conjecture 1: Tutte's $5$-flow conjecture
  • Conjecture 2
  • Conjecture 3
  • Theorem 1
  • Lemma 2
  • Theorem 3
  • Lemma 5: thom2
  • Lemma 6
  • Proposition 1: diestel2017graph, Corollary 6.3.2
  • Lemma 8
  • ...and 38 more