Jaeger-type orientations of random regular graphs

TL;DR

This paper investigates the existence of specific orientations in random regular graphs, extending Jaeger's conjecture, and provides probabilistic results for various parameters using advanced combinatorial methods.

Contribution

It proves that certain $p$-orientations exist with high probability in random $d$-regular graphs for multiple $(d,p)$ pairs, including some cases related to Jaeger's conjecture.

Findings

High probability existence of $p$-orientations for several $(d,p)$ pairs.

Counterexamples for Jaeger's conjecture in specific cases.

Connection established between orientations and maximum bisection size.

Abstract

We consider $p$-orientations, which are defined to be orientations of $d$-regular graphs such that every vertex either has in-degree $p$ or out-degree $p$. These generalise the orientations considered in Jaeger's conjecture, where $d=4p+1$. Working with random $d$-regular graphs using the small subgraph conditioning method, we prove that a $d$-regular graph has a $p$-orientation with high probability for several values of $(d,p)$, including the $p=3,4$ cases of Jaeger's conjecture (known to be deterministically false). Some negative results are obtained by exploiting a connection with maximum bisection size.