Manhattan and Chebyshev flows

TL;DR

This paper explores multidimensional nowhere-zero flows in graphs using Manhattan and Chebyshev norms, introduces new flow numbers, bounds, structural insights, and proposes conjectures related to famous flow conjectures.

Contribution

It extends flow theory by defining new flow numbers based on Manhattan and Chebyshev norms, and introduces the concept of t-flow-pairs with conjectures surpassing Tutte's 5-flow conjecture.

Findings

Flow numbers are always rational.

In two dimensions, flow numbers distinguish cubic graphs by 3-edge-colorability.

The two flow numbers are equal for any bridgeless graph.

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 $\Phi_d^1(G)$ and $\Phi_d^\infty(G)$, 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 $\Phi^1_2(G)$ and $\Phi^\infty_2(G)$ 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.