Packing mixed hyperarborescences

TL;DR

This paper introduces a new orientation theorem for packing mixed hyperarborescences, simplifying previous proofs and extending results to hypergraphs with specific packing constraints.

Contribution

It provides a new orientation theorem for mixed hyperarborescences and extends existing results to hypergraphs with bounded packing and root constraints.

Findings

Simplified proof of Gao and Yang's result using orientation theorem

Extended packing theorem to mixed hypergraphs with vertex-specific bounds

Established conditions for packing a specified number of hyperarborescences

Abstract

The aim of this paper is twofold. We first provide a new orientation theorem which gives a natural and simple proof of a result of Gao, Yang \cite{GY} on matroid-reachability-based packing of mixed arborescences in mixed graphs by reducing it to the corresponding theorem of Cs. Kir\'aly \cite{cskir} on directed graphs. Moreover, we extend another result of Gao, Yang \cite{GY2} by providing a new theorem on mixed hypergraphs having a packing of mixed hyperarborescences such that their number is at least $\ell$ and at most $\ell'$, each vertex belongs to exactly $k$ of them, and each vertex $v$ is the root of least $f(v)$ and at most $g(v)$ of them.