AI 中文总结
在图论与群论交叉领域，寻找顶点传递图和有向图中的哈密顿圈是经典问题。本文构造无限多个最长有向圈遗漏至少\(n/12\)个顶点的连通顶点传递有向图，证实相关猜想，还将有向圈长度下限从\(\Omega(n^{1/3})\)提高到\(\Omega(\sqrt n)\) 。
AI 中文摘要
在图论和群论的交叉领域，寻找顶点传递图和有向图中的哈密顿圈是一个经典问题。在无向情况下，这可追溯到Lovász和Thomassen的著名猜想，即每个足够大的连通顶点传递图都是哈密顿图。有向类似问题历史更丰富，始于1946年的Rankin。1978年Trotter和Erdős表明连通顶点传递有向图不一定是哈密顿图。1981年Alspach提出问题，最近Bucić等人构造出最长有向圈遗漏\((1 - o(1))\log n\)个顶点的连通顶点传递有向图，并猜想遗漏顶点数可线性增长。本文通过构造无限多个最长有向圈遗漏至少\(n/12\)个顶点的连通顶点传递有向图，证实了他们的猜想。同时，Bucić等人证明了\(n\)个顶点的连通顶点传递有向图包含长度为\(\Omega(n^{1/3})\)的有向圈，本文将此下限提高到\(\Omega(\sqrt n)\)，与1979年Babai关于无向顶点传递图的经典定理的阶数相匹配。
英文摘要
The search for Hamiltonian cycles in vertex-transitive graphs and digraphs is a classical problem at the interface of graph theory and group theory. In the undirected setting, this goes back to the well-known conjectures of Lovász and Thomassen concerning Hamiltonian paths and cycles in connected vertex-transitive graphs. Dating back to Rankin's 1946 work, the directed analogue has an even longer history, linking the search for long cycles to classical group-rearrangement problems. Trotter and Erdős showed in 1978 that connected vertex-transitive digraphs need not be Hamiltonian. In light of this result, Alspach asked in 1981 whether there exist connected vertex-transitive digraphs whose longest directed cycle misses arbitrarily many vertices. This question was only recently resolved by Bucić, Hendrey, Mohar, Steiner and Yepremyan, who constructed connected vertex-transitive digraphs on $n$ vertices whose longest directed cycle omits $(1-o(1))\log n$ vertices. They conjectured that the number of omitted vertices can grow linearly with $n$, remarking that it would already be interesting to improve their logarithmic lower bound to a polynomial bound. In this paper, we confirm their conjecture in a strong form by constructing infinitely many connected vertex-transitive digraphs on $n$ vertices whose longest directed cycle omits at least $n/12$ vertices. In the same work, Bucić, Hendrey, Mohar, Steiner and Yepremyan also proved that every connected vertex-transitive digraph on $n$ vertices contains a directed cycle of length $Ω(n^{1/3})$, giving the first lower bound for this problem that grows with $n$. We improve this to $Ω(\sqrt n)$, matching the order of Babai's classical theorem from 1979 for undirected vertex-transitive graphs.
Comments14 pages
