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Manhattan and Chebyshev flows

Lukáš Gáborik, Sascha Kurz, Giuseppe Mazzuoccolo, Jozef Rajník, Florian Rieg

TL;DR

This work extends multidimensional nowhere-zero flows to non-Euclidean geometries by introducing Manhattan and Chebyshev norm flows in $d$ dimensions, defining $\Phi_d^1(G)$ and $\Phi_d^\infty(G)$ as their flow numbers. The authors prove rationality of these invariants for all bridgeless graphs and show the two-dimensional case satisfies $\Phi_2^1(G)=\Phi_2^\infty(G)$, linking them to cycle covers and $t$-flow-pairs. They derive general upper bounds via dimension and construct Hadamard-matrix-based flows to achieve tight bounds in certain families, while also establishing a sharp 2D bound for cubic graphs and a lower bound for non-$3$-edge-colorable graphs. The introduction of $t$-flow-pairs connects to Seymour's $6$-flow theorem and Tutte's $5$-flow conjecture, with computational evidence supporting the existence of half-flow-pairs in many snarks and several conjectures that aim to strengthen classical flow results. Altogether, the paper provides a rational, structurally rich framework that links norms, cycle covers, and longstanding flow conjectures, and it proposes concrete avenues for progress toward Tutte-type bounds via non-Euclidean flows.

Abstract

We investigate multidimensional nowhere-zero flows of bridgeless graphs. By extending the established use of the Euclidean norm, this paper considers the Manhattan and Chebyshev norms, leading to the definition of the flow numbers $Φ_d^1(G)$ and $Φ_d^\infty(G)$, respectively. These flow numbers are always rational and in two dimensions, they distinguish between cubic graphs that are 3-edge-colourable and those that are not. We also prove that, for any bridgeless graph $G$, the two values $Φ^1_2(G)$ and $Φ^\infty_2(G)$ are the same. We give new upper and lower bounds and structural results, and we find connections with cycle covers. Finally, we introduce the idea of $t$-flow-pairs, which comes from a method used in Seymour's proof of the 6-flow theorem, and we propose new conjectures that could be stronger than Tutte's famous 5-flow conjecture.

Manhattan and Chebyshev flows

TL;DR

This work extends multidimensional nowhere-zero flows to non-Euclidean geometries by introducing Manhattan and Chebyshev norm flows in dimensions, defining and as their flow numbers. The authors prove rationality of these invariants for all bridgeless graphs and show the two-dimensional case satisfies , linking them to cycle covers and -flow-pairs. They derive general upper bounds via dimension and construct Hadamard-matrix-based flows to achieve tight bounds in certain families, while also establishing a sharp 2D bound for cubic graphs and a lower bound for non--edge-colorable graphs. The introduction of -flow-pairs connects to Seymour's -flow theorem and Tutte's -flow conjecture, with computational evidence supporting the existence of half-flow-pairs in many snarks and several conjectures that aim to strengthen classical flow results. Altogether, the paper provides a rational, structurally rich framework that links norms, cycle covers, and longstanding flow conjectures, and it proposes concrete avenues for progress toward Tutte-type bounds via non-Euclidean flows.

Abstract

We investigate multidimensional nowhere-zero flows of bridgeless graphs. By extending the established use of the Euclidean norm, this paper considers the Manhattan and Chebyshev norms, leading to the definition of the flow numbers and , respectively. These flow numbers are always rational and in two dimensions, they distinguish between cubic graphs that are 3-edge-colourable and those that are not. We also prove that, for any bridgeless graph , the two values and are the same. We give new upper and lower bounds and structural results, and we find connections with cycle covers. Finally, we introduce the idea of -flow-pairs, which comes from a method used in Seymour's proof of the 6-flow theorem, and we propose new conjectures that could be stronger than Tutte's famous 5-flow conjecture.
Paper Structure (8 sections, 19 theorems, 20 equations, 2 figures, 2 tables)

This paper contains 8 sections, 19 theorems, 20 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Rationality of flow numbers
  3. General upper bounds by dimension
  4. Manhattan flows via Hadamard matrices
  5. Two-dimensional case
  6. t-flow-pairs
  7. Conjectures and open problems
  8. Acknowledgements

Key Result

Theorem 3

For each bridgeless graph $G$ and each integer $d \ge 1$, the values $\Phi_d^\infty(G)$ and $\Phi_d^1(G)$ are rational.

Figures (2)

  • Figure 1: The annulus in the Manhattan norm (left) with respect to the annulus in the Chebyshev norm (right)
  • Figure 2: A $(\frac{5}{2},2)$-ChNZF of the Petersen graph.

Theorems & Definitions (52)

  • Conjecture 1
  • Conjecture 2
  • Theorem 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Corollary 6
  • Proposition 7
  • ...and 42 more