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High-Dimensional $p$-Normed Flows

Chenxing Li, Jiaao Li, Rong Luo, Bo Su

TL;DR

The paper extends flow theory by defining and analyzing $d$-dimensional $p$-normed nowhere-zero flows and the corresponding flow index $\phi_{d,p}(G)$. It introduces the $\Omega$-flow method to derive universal upper bounds, obtaining $\phi_{2,p}(G)\le 3$ and $\phi_{3,p}(G)\le 1+2^{1/p}$ (with $\phi_{3,\infty}(G)=2$ and $\phi_{3,1}(G)\le 9/4$), and shows that graphs with oriented cycle covers achieve $\phi_{d,p}(G)=2$ for suitable $d$. The results connect to cycle covers, provide sharp bounds for graphs with a $4$-NZF, and tie into classical conjectures such as Tutte's $5$-flow and Jain's $S^2$-Flow, while outlining directions for duality, monotonicity, and the $S^2_p$-Flow program. The work synthesizes combinatorial, geometric, and topological viewpoints, offering a versatile framework for future exploration of normed vector flows in graphs.

Abstract

We generalize Tutte's integer flows and the $d$-dimensional Euclidean flows of Mattiolo, Mazzuoccolo, Rajník, and Tabarelli to \emph{$d$-dimensional $p$-normed nowhere-zero flows} and define the corresponding flow index $φ_{d,p}(G)$ to be the infimum over all real numbers $r$ for which $G$ admits a $d$-dimensional $p$-normed nowhere-zero $r$-flow. For any bridgeless graph $G$ and any $p\ge 1$, we establish general upper bounds, including $φ_{2,p}(G) \le 3$, $φ_{3,p}(G) \le 1+\sqrt{2}$, and tight bounds for graphs admitting a $4$-NZF. For graphs with oriented $(k+1)$-cycle $2l$-covers, we show that $φ_{k,p}(G) = 2$, which implies $φ_{2,p}(G) = 2$ for graphs admitting a nowhere-zero $3$-flow and $φ_{3,p}(G) = 2$ for those admitting a nowhere-zero $4$-flow. These results extend classical flow theory to arbitrary norms, provide supporting evidences for Tutte's $5$-flow Conjecture and Jain's $S^2$-Flow Conjecture, and connect combinatorial flows with geometric and topological perspectives.

High-Dimensional $p$-Normed Flows

TL;DR

The paper extends flow theory by defining and analyzing -dimensional -normed nowhere-zero flows and the corresponding flow index . It introduces the -flow method to derive universal upper bounds, obtaining and (with and ), and shows that graphs with oriented cycle covers achieve for suitable . The results connect to cycle covers, provide sharp bounds for graphs with a -NZF, and tie into classical conjectures such as Tutte's -flow and Jain's -Flow, while outlining directions for duality, monotonicity, and the -Flow program. The work synthesizes combinatorial, geometric, and topological viewpoints, offering a versatile framework for future exploration of normed vector flows in graphs.

Abstract

We generalize Tutte's integer flows and the -dimensional Euclidean flows of Mattiolo, Mazzuoccolo, Rajník, and Tabarelli to \emph{-dimensional -normed nowhere-zero flows} and define the corresponding flow index to be the infimum over all real numbers for which admits a -dimensional -normed nowhere-zero -flow. For any bridgeless graph and any , we establish general upper bounds, including , , and tight bounds for graphs admitting a -NZF. For graphs with oriented -cycle -covers, we show that , which implies for graphs admitting a nowhere-zero -flow and for those admitting a nowhere-zero -flow. These results extend classical flow theory to arbitrary norms, provide supporting evidences for Tutte's -flow Conjecture and Jain's -Flow Conjecture, and connect combinatorial flows with geometric and topological perspectives.
Paper Structure (12 sections, 17 theorems, 93 equations, 1 figure, 1 table)

This paper contains 12 sections, 17 theorems, 93 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Background, motivation, and definitions
  3. Main results: upper bounds for flow indices
  4. Main results: connections with cycle covers
  5. A general method for finding upper bounds of flow indices
  6. Upper bounds for general $p$-norms
  7. Proof of Theorem \ref{['thm:general-2-3-bounds']}
  8. Proof of Theorem \ref{['thm:3-dim-p-12-flow']}
  9. Upper bounds under oriented $k$-cycle $2l$-covers
  10. Discussions and Open Problems
  11. Remarks on Theorem \ref{['thm:3-dim-p-12-flow']}
  12. Open problems and conjectures

Key Result

Theorem 1.6

Let $G$ be a bridgeless graph. Then we have

Figures (1)

  • Figure 1: Curves of the functions $g_1(p)$, $g_2(p)$, and $g_3(p)$.

Theorems & Definitions (35)

  • Definition 1.1
  • Definition 1.2
  • Conjecture 1.3: Tutte’s $5$-Flow Conjecture
  • Conjecture 1.4: Jain’s $S^2$-Flow Conjecture
  • Conjecture 1.5: Jain's $S^2$-Map Conjecture
  • Theorem 1.6: Mattiolo et al. mattiolo2023d
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 25 more