# Counting degree-constrained subgraphs and orientations

## TL;DR

This paper introduces gauge transformations as a novel combinatorial proof technique and applies it to establish lower bounds on the number of Eulerian orientations and subgraphs in regular graphs.

## Contribution

It provides a new proof of Schrijver's theorem using gauge transformations and relates Eulerian orientations to regular subgraphs.

## Key findings

- Lower bound on Eulerian orientations in regular graphs
- Comparison between Eulerian orientations and regular subgraphs
- Application of gauge transformations in combinatorics

## Abstract

The goal of this short paper to advertise the method of gauge transformations (aka holographic reduction, reparametrization) that is well-known in statistical physics and computer science, but less known in combinatorics. As an application of it we give a new proof of a theorem of A. Schrijver asserting that the number of Eulerian orientations of a $d$--regular graph on $n$ vertices with even $d$ is at least $\left(\frac{\binom{d}{d/2}}{2^{d/2}}\right)^n$. We also show that a $d$--regular graph with even $d$ has always at least as many Eulerian orientations as $(d/2)$--regular subgraphs.