发表机构
University of Illinois Urbana–Champaign（伊利诺伊大学厄巴纳-香槟分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文改进了 Lovász 猜想的已知下界，证明每个连通的顶点传递图都包含一条长度为 $n^{1-o(1)}$ 的路径，方法结合了结构划分、生成树遍历和局部引理。
AI 中文摘要
Lovász 于 1969 年提出的著名猜想询问是否每个连通的顶点传递图都有哈密顿路径。Bucić、Christoph、Pokrovskiy 和 Steiner 最近证明了每个包含 $n$ 个顶点的此类图都含有一条长度为 $n^{2/3-o(1)}$ 的环。在本文中，我们将此界改进为 $n^{1-o(1)}$。我们的证明使用 Tessera 和 Tointon 的结构定理，首先获得原始图中顶点集的一个划分，使得各部分在原始图中具有小直径。当各部分较大时，我们反复遍历商图中最大度至多为三的生成树，并使用 Lovász 局部引理沿此遍历连接随机短路径，从而在原始图中提取一条长路径。当各部分较小时，我们应用 Babai 的收缩引理将问题简化为在具有有界生成元数和有界幂零类的幂零群的连通 Cayley 图中寻找长路径，然后证明这样的包含 $m$ 个顶点的 Cayley 图包含一条长度为 $m^{1-o(1)}$ 的路径。
英文摘要
The celebrated conjecture of Lovász from 1969 asks whether every connected vertex-transitive graph has a Hamiltonian path. Bucić, Christoph, Pokrovskiy and Steiner recently proved that every such graph on $n$ vertices contains a cycle of length $n^{2/3-o(1)}$. In this paper, we improve this bound to $n^{1-o(1)}$. Our proof uses a structure theorem of Tessera and Tointon to first obtain a partition of the vertex set into sets of small diameter in the original graph. When the parts are large, we repeatedly traverse a spanning tree of maximum degree at most three in the quotient graph, and use the Lovász local lemma to join random short paths along this traversal and extract a long path in the original graph. When the parts are small, we apply Babai's contraction lemma to reduce the problem to finding a long path in a connected Cayley graph of a nilpotent group with boundedly many generators and bounded nilpotency class, and then show that such a Cayley graph on $m$ vertices contains a path on $m^{1-o(1)}$ vertices.
Comments21 pages; comments welcome!
