关于不同长度不交有向圈的Lichiardopol猜想的证明
Proof of Lichiardopol's conjecture on disjoint directed cycles of distinct lengths
AI总结：
本文基于结构有向图理论的相关技术，完整证明了Lichiardopol关于最小出度足够大的有向图含不同长度不交有向圈的猜想，并将结果推广到带权情形。
AI中文摘要：
研究具有大最小出度的有向图中可保证何种结构的相关问题十分有趣，这类问题常表述简洁却极难处理，著名例子是2014年提出的Lichiardopol猜想：存在函数$g:\mathbb{N}\rightarrow \mathbb{N}$，使得每个最小出度至少为$g(k)$的有向图都包含$k$个顶点不交、长度不同的有向圈。本文基于第二作者的前期工作，完整证明了该猜想，并将结果推广到带权情形。证明结合了结构有向图理论的诸多要素，包括蝴蝶子式、有向缠结、Robertson和Seymour提出的缠结-墙定理的有向类似物，以及Giannopoulou、Kawarabayashi、Kreutzer和Kwon提出的有向平墙定理的局部变体。这些在最小度条件研究中略显非典型的技术具有独立研究价值，或可应用于更多领域。
英文摘要：
There is a fascinating array of interrelated questions studying which structures can be guaranteed in digraphs of large minimum out-degree. These often have intriguingly simple statements, yet seem surprisingly difficult to approach. A well-known example is Lichiardopol's conjecture (2014), stating that there exists a function $g:\mathbb{N}\rightarrow \mathbb{N}$ such that every digraph with minimum out-degree at least $g(k)$ contains $k$ vertex-disjoint directed cycles of distinct lengths. In this paper, building on earlier work of the second author, we confirm this conjecture in full generality. We also generalise this result to a weighted setting. Our proof uses and combines many ingredients from structural digraph theory such as butterfly minors, directed tangles, a directed analogue of the Tangle-Wall Theorem due to Robertson and Seymour as well as a local variant of the Directed Flat Wall Theorem due to Giannopoulou, Kawarabayashi, Kreutzer and Kwon. These techniques, which are somewhat atypical in the study of minimum degree conditions, may be of independent interest and may find further applications.
