Almost all 9-regular graphs have a modulo-5 orientation

TL;DR

This paper proves that almost all 9-regular graphs can be oriented to satisfy a specific in-degree condition, supporting a broader conjecture about regular graphs and edge orientations.

Contribution

The paper demonstrates that the conjecture holds asymptotically almost surely for random 9-regular graphs, extending previous work for p=1 to p=2.

Findings

Almost all 9-regular graphs have a modulo-5 orientation.

The conjecture holds asymptotically almost surely for these graphs.

Results utilize the small subgraph conditioning method.

Abstract

In 1972 Tutte famously conjectured that every 4-edge-connected graph has a nowhere zero 3-flow; this is known to be equivalent to every 5-regular, 4-edge-connected graph having an edge orientation in which every in-degree is either 1 or 4. Jaeger conjectured a generalization of Tutte's conjecture, namely, that every $4p+1$-regular, $4p$-edge-connected graph has an edge orientation in which every in-degree is either $p$ or $3p+1$. Inspired by the work of Pralat and Wormald investigating $p=1$, for $p=2$ we show this holds asymptotically almost surely for random 9-regular graphs. It follows that the conjecture holds for almost all 9-regular, 8-edge-connected graphs. These results make use of the technical small subgraph conditioning method.