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Geometric description of $d$-dimensional flows of a graph

Davide Mattiolo, Giuseppe Mazzuoccolo, Jozef Rajník, Gloria Tabarelli

TL;DR

The paper generalizes nowhere-zero flows to $d$-dimensional vector flows and introduces the $d$-dimensional flow number $φ_d(G)$, aiming to geometrically characterize such flows and connect them to cycle double covers. It defines the sets $\oldsymbol{Σ}_d$ and $H_d$ and proves that a graph admits an $H_d$-flow if and only if it has an oriented $d$-cycle double cover, establishing a constructive correspondence between flows and oriented covers. A key contribution is the realization of $H_d$ as the vertex set of the line graph of the crown graph $Cr(2d)$, enabling a concrete geometric and graph-structural interpretation. The paper also derives upper bounds on $φ_{d-1}(G)$ and $φ_{d-2}(G)$ under various cycle double-cover assumptions, and discusses consequences such as $φ_{d-1}(G)=2$ when an oriented $d$-CDCC exists, connecting to known conjectures and special cases like $d=3$ and the $S^{d-2}$-flow framework.

Abstract

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 $\mathbb{R}^d$ whose (Euclidean) norm lies in the interval $[1, r-1]$. Such a notion is a natural generalization of the well-known concept of a circular nowhere-zero $r$-flow (i.e.\ $d = 1$). The minimum of the real numbers $r$ such that a graph $G$ admits an $(r, d)$-NZF is called the $d$-dimensional flow number of $G$ and is denoted by $φ_d(G)$. In this paper we provide a geometric description of some $d$-dimensional flows on a graph $G$, and we prove that the existence of a suitable cycle double cover of $G$ is equivalent, for $G$, to admit such a geometrically constructed $(r,d)$-NZF. This geometric approach allows us to provide upper bounds for $φ_{d-2}(G)$ and $φ_{d-1}(G)$, assuming that $G$ admits an (oriented) $d$-cycle double cover.

Geometric description of $d$-dimensional flows of a graph

TL;DR

The paper generalizes nowhere-zero flows to -dimensional vector flows and introduces the -dimensional flow number , aiming to geometrically characterize such flows and connect them to cycle double covers. It defines the sets and and proves that a graph admits an -flow if and only if it has an oriented -cycle double cover, establishing a constructive correspondence between flows and oriented covers. A key contribution is the realization of as the vertex set of the line graph of the crown graph , enabling a concrete geometric and graph-structural interpretation. The paper also derives upper bounds on and under various cycle double-cover assumptions, and discusses consequences such as when an oriented -CDCC exists, connecting to known conjectures and special cases like and the -flow framework.

Abstract

A -dimensional nowhere-zero -flow on a graph , an -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 a circular nowhere-zero -flow (i.e.\ ). The minimum of the real numbers such that a graph admits an -NZF is called the -dimensional flow number of and is denoted by . In this paper we provide a geometric description of some -dimensional flows on a graph , and we prove that the existence of a suitable cycle double cover of is equivalent, for , to admit such a geometrically constructed -NZF. This geometric approach allows us to provide upper bounds for and , assuming that admits an (oriented) -cycle double cover.
Paper Structure (4 sections, 7 theorems, 3 equations, 1 figure)

This paper contains 4 sections, 7 theorems, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Geometric description of a $d$-dimensional flow
  3. A description of $H_d$ as vertex set of a line graph
  4. Upper bounds for $\phi_{d-1}(G)$ and $\phi_{d-2}(G)$

Key Result

Proposition 3

Let $G$ be a graph. Then (a) and (b) below are equivalent, and they imply (c) where $G$ has a nowhere zero 3-flow. $G$ has an $R_3$-flow. $G$ has an $S^1$-flow. If $G$ is cubic the three statements are equivalent, and $G$ satisfies (a), (b), (c) if and only if $G$ is bipartite.

Figures (1)

  • Figure 1: An $S^2$-flow on the Petersen Graph.

Theorems & Definitions (14)

  • Conjecture 1: $S^2$-flow Conjecture
  • Conjecture 2: Oriented $5$-Cycle Double Cover Conjecture
  • Proposition 3
  • Remark 1
  • Theorem 4
  • proof
  • Corollary 5
  • Proposition 6
  • proof
  • Proposition 7
  • ...and 4 more