Almost All 9-Regular Graphs Have a Modulo-5 Orientation

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 nowhere-zero 3-flow 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 Prałat and Wormald investigating $p=1$, we address $p=2$ to show that the conjecture 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.