扩充超图使其具有基于拟阵的$(f,g)$-有界$(α,β)$-限制根超树填充
Augmenting a hypergraph to have a matroid-based $(f,g)$-bounded $(α,β)$-limited packing of rooted hypertrees
AI总结：
本文通过改编证明扩展图中根树填充理论，刻画满足顶点根树数量及总根数量上下界的基于拟阵的填充存在性，解决超图版本及扩充问题，所提方法具潜在应用价值。
摘要：
本文旨在进一步发展图中树填充的理论。首先，我们通过改编Lovász对Edmonds关于有向图中生成树形图填充的开创性结果的证明，证明了Nash-Williams和Tutte关于生成树填充的经典结果。我们关于图的主要结果通过刻画满足以下附加条件的此类填充的存在性，扩展了Katoh和Tanigawa关于基于拟阵的根树填充的定理：对于每个顶点$v$，以$v$为根的树的数量有下界$f(v)$和上界$g(v)$，且根的总数有下界$α$和上界$β$。我们还回答了该问题的超图版本。此外，我们能够解决后者的扩充版本问题，即添加最少数量的边以获得此类填充。本文为解决这些问题所开发的方法未来可能有其他应用。
英文摘要：
The aim of this paper is to further develop the theory of packing trees in a graph. We first prove the classic result of Nash-Williams \cite{NW} and Tutte \cite{Tu} on packing spanning trees by adapting Lovász' proof \cite{Lov} of the seminal result of Edmonds \cite{Egy} on packing spanning arborescences in a digraph. Our main result on graphs extends the theorem of Katoh and Tanigawa \cite{KT} on matroid-based packing of rooted trees by characterizing the existence of such a packing satisfying the following further conditions: for every vertex $v$, there are a lower bound $f(v)$ and an upper bound $g(v)$ on the number of trees rooted at $v$ and there are a lower bound $α$ and an upper bound $β$ on the total number of roots. We also answer the hypergraphic version of the problem. Furthermore, we are able to solve the augmentation version of the latter problem, where the goal is to add a minimum number of edges to have such a packing. The methods developed in this paper to solve these problems may have other applications in the future.
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