Exponentially many nowhere-zero $Z_3$-, $Z_4$-, and $Z_6$-flows

TL;DR

This paper demonstrates that graphs have exponentially many nowhere-zero flows over Z_3, Z_4, and Z_6, providing a counting perspective complementing existence proofs and introducing a new splitting lemma for 6-edge-connected graphs.

Contribution

It establishes exponential lower bounds on the number of nowhere-zero flows in graphs and introduces a novel splitting lemma for 6-edge-connected graphs.

Findings

Graphs have exponentially many nowhere-zero Z_3-, Z_4-, and Z_6-flows.

A new splitting lemma for 6-edge-connected graphs is developed.

Results offer a counting approach as an alternative to existence proofs.

Abstract

We prove that, in several settings, a graph has exponentially many nowhere-zero flows. These results may be seen as a counting alternative to the well-known proofs of existence of $Z_3$-, $Z_4$-, and $Z_6$-flows. In the dual setting, proving exponential number of 3-colorings of planar triangle-free graphs is a related open question due to Thomassen. As a part of the proof we obtain a new splitting lemma for 6-edge-connected graphs, that may be of independent interest.