Equitable orientations of sparse uniform hypergraphs

TL;DR

This paper proves a conjecture regarding the existence of equitable orientations in sparse uniform hypergraphs, ensuring indegree and outdegree are balanced, advancing understanding in hypergraph orientation theory.

Contribution

It provides a proof for a conjecture on the existence of equitable orientations in sparse r-uniform hypergraphs, a problem previously unresolved.

Findings

Confirmed the existence of equitable orientations in sparse hypergraphs

Extended the theory of hypergraph orientations to new classes of hypergraphs

Provided a constructive proof for the conjecture

Abstract

Caro, West, and Yuster studied how $r$-uniform hypergraphs can be oriented in such a way that (generalizations of) indegree and outdegree are as close to each other as can be hoped. They conjectured an existence result of such orientations for sparse hypergraphs, of which we present a proof.