# Covers, orientations and factors

## TL;DR

This paper provides a new proof for the inequality relating Eulerian orientations and half graphs in Eulerian graphs, and explores identities and inequalities for these counts in graph covers.

## Contribution

It offers a simple new proof of a known inequality and extends the analysis to identities and inequalities for graph covers.

## Key findings

- Proved that $ ext{ε}(G) ext{geq} h(G)$ for Eulerian graphs with equality iff bipartite.
- Derived identities for Eulerian orientations and half graphs in 2-covers.
- Established inequalities relating these quantities in graph covers.

## Abstract

Given a graph $G$ with only even degrees let $\varepsilon(G)$ denote the number of Eulerian orientations, and let $h(G)$ denote the number of half graphs, that is, subgraphs $F$ such that $d_F(v)=d_G(v)/2$ for each vertex $v$. Recently, Borb\'enyi and Csikv\'ari proved that $\varepsilon(G)\geq h(G)$ holds true for all Eulerian graphs with equality if and and only if $G$ is bipartite. In this paper we give a simple new proof of this fact, and we give identities and inequalities for the number of Eulerian orientations and half graphs of a $2$-cover of a graph $G$.