# Modulo orientations with bounded out-degrees

## TL;DR

This paper proves that highly edge-connected graphs and graphs with many spanning trees can be oriented to have balanced out-degrees within certain bounds, extending understanding of graph orientations with degree constraints.

## Contribution

It establishes new conditions under which graphs can be oriented to achieve bounded out-degrees, linking edge-connectivity and spanning trees to orientation properties.

## Key findings

- Graphs with (3k-3)-edge-connectivity admit orientations with bounded out-degree deviations.
- Graphs with 2k-2 edge-disjoint spanning trees can also be oriented with bounded out-degree, with stricter bounds.
- The results generalize and strengthen previous orientation theorems based on connectivity and spanning tree conditions.

## Abstract

Let $G$ be a graph, let $k$ be a positive integer, and let $p:V(G)\rightarrow Z_k$ be a mapping with $|E(G)| \stackrel{k}{\equiv}\sum_{v\in V(G)}p(v) $. In this paper, we show that if $G$ is $(3k-3)$-edge-connected, then it has an orientation such that for each vertex $v$, $|d^+_G(v)-d_G(v)/2| < k$; also if $G$ contains $2k-2$ edge-disjoint spanning trees, then it admits such an orientation but by imposing greater out-degree bounds.