Almost all 5-regular graphs have a 3-flow

TL;DR

This paper proves that almost all 5-regular, 4-edge-connected graphs have a specific edge orientation related to Tutte's 3-flow conjecture, confirming the conjecture's validity for these graphs asymptotically.

Contribution

It demonstrates that Tutte's 3-flow conjecture holds asymptotically almost surely for random 5-regular graphs, extending the conjecture's validity to a broad class of graphs.

Findings

The conjecture holds asymptotically almost surely for random 5-regular graphs.

Almost all 4-edge connected 5-regular graphs satisfy the conjecture.

The result confirms the conjecture for a large class of graphs.

Abstract

Tutte conjectured in 1972 that every 4-edge connected graph has a nowhere-zero 3-flow. This has long been known to be equivalent to the conjecture that every 5-regular 4-edge-connected graph has an edge orientation in which every out-degree is either 1 or 4. We show that the assertion of the conjecture holds asymptotically almost surely for random 5-regular graphs. It follows that the conjecture holds for almost all 4-edge connected 5-regular graphs.