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

TL;DR

The paper proves that a random $9$-regular graph is asymptotically almost surely modulo-$5$ orientable, establishing Jaeger’s conjecture for the case $p=2$ via the small subgraph conditioning method. It works in the Bollobás pairing model $\mathcal{P}_{n,9}$, defines $Y$ as the count of valid orientations, and computes the first and second moments, showing $\mathbb{E}[Y]\sim 3\left(\frac{81}{8}\right)^{n/2}$ and $\mathbb{E}[Y(Y-1)]\sim\left(\frac{81}{8}\right)^n\frac{81}{7}$; the naive second-moment bound fails, necessitating SSCM. The analysis derives a unique interior maximum for the exponent, analyzes boundary behavior, and proves that a limiting Gaussian-type contribution yields $\mathbb{P}(Y>0)\to 1$, thereby confirming the existence of a modulo-$5$ orientation a.a.s. The results highlight the power of small subgraph conditioning in high-regularity random graphs and bolster the asymptotic validity of Jaeger’s conjecture for $p=2$.

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.

Theorems & Definitions (15)

• Conjecture 1
• Conjecture 2
• Conjecture 3
• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Corollary 2.3
• Theorem 2.4: Small Subgraph Conditioning Method
• Lemma 2.5
• Corollary 2.6