发表机构
Miami University（迈阿密大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过修改 Häggkvist 和 Thomason 的证明，给出了 Taylor 定理（关于鲁棒外出扩张图包含所有定向哈密顿圈）的免正则引理证明，并推广到哈密顿连通性与链结，应用替代正则引理于最小半度或总度条件下的有向图结果。
AI 中文摘要
Häggkvist 和 Thomason 证明了每个最小半度至少为 $(\ rac{5}{12}+o(1))n$ 的 $n$ 顶点有向图包含哈密顿圈的每一种定向。Kelly 后来使用 Szemerédi 正则引理将其改进为渐近最优的界 $(\ rac{3}{8}+o(1))n$。Taylor 随后推广了 Kelly 的定理，证明了每个具有线性最小半度的足够大的鲁棒外出扩张图包含哈密顿圈的每一种定向。我们重新审视 Häggkvist 和 Thomason 的证明，并表明它可以被修改以给出 Taylor 定理（从而 Kelly 定理）的免正则引理证明。在此基础上，我们还给出了关于鲁棒外出扩张图中哈密顿连通性和链结的相关结果的免正则引理证明。作为这些结果的应用，我们能够在已知结果中用正则引理替换为：在最小半度至少为 $\ rac{n}{2}$ 的 $n$ 顶点有向图中，以及在最小总度至少为 $(1+o(1))n$ 的 $n$ 顶点有向图中，任意定向的哈密顿圈。第一步是证明，在一个具有线性最小半度的 $n$ 顶点鲁棒外出扩张图中，两个均匀选择的、大小为对数的互不相交集合以高概率满足 Hall 条件。第二步是证明鲁棒扩张性质被均匀选择的线性大小集合所继承。在这两种情况下，我们使用 Kleitman--Winston 和 Sapozhenko 的图容器方法将可能的障碍减少到足够小的族，以便进行联合界。
英文摘要
Häggkvist and Thomason proved that every $n$-vertex oriented graph with minimum semidegree at least $(\frac{5}{12}+o(1))n$ contains every orientation of a Hamilton cycle. Using Szemerédi's regularity lemma, Kelly later improved this to the asymptotically sharp bound $(\frac{3}{8}+o(1))n$. Taylor subsequently generalized Kelly's theorem by proving that every sufficiently large robust outexpander with linear minimum semidegree contains every orientation of a Hamilton cycle. We revisit Häggkvist and Thomason's proof and show that it can be modified to give a regularity-free proof of Taylor's theorem (and thus Kelly's theorem). Building on this, we also give regularity-free proofs of related results on Hamilton-connectivity and linkage in robust outexpanders. As applications of these results, we are able to replace the use of the regularity lemma in known results on arbitrary orientations of Hamilton cycles in $n$-vertex digraphs with minimum semidegree at least $\frac{n}{2}$ and in $n$-vertex digraphs with minimum total degree at least $(1+o(1))n$. The first step is to show that, in an $n$-vertex robust outexpander with linear minimum semidegree, two uniformly chosen disjoint sets of logarithmic size satisfy Hall's condition with high probability. The second is to show that robust expansion is inherited by uniformly chosen linear sized sets. In both cases, we use the graph-container methods of Kleitman--Winston and Sapozhenko to reduce the possible obstructions to a small enough family to permit a union bound.
Comments28 pages (plus 7-page appendix); 4 figures
