有限宽度有向六角网格的精确端度阈值
The Exact End-Degree Threshold for Finite-Width Directed Hexagonal Grids
AI总结：
该研究解决了Hamann和Heuer提出的问题，确定了有限宽度有向六角网格的精确端度阈值k(n)，并得出对偶极值参数W(d)的表达式。
AI中文摘要：
网格定理为无限图和有向图中端点结构与类网格子图的存在性提供了基本关联。对每个固定宽度n，设k(n)为最小正整数，使得每个具有入度至少为k(n)的端点的有向图，都包含宽度为n的有向六角网格的细分图。Hamann和Heuer提出了k(n)的精确值问题。我们通过证明k(n)=⌊3n/2⌋−1（对所有n≥4），且k(1)=1、k(2)=2、k(3)=4，解决了该问题。对于上界，我们建立了带标记令牌滑动的有限空缺定理，将其转换为辅助射线有向图上的闭合有向调度，并通过全新的干净链接提升该调度；对于匹配下界，我们使用射线与三臂树乘积的交替定向，并证明了不相交有向路径的无通行性质。我们还确定了对偶极值参数：设W(d)为入度为d的端点（所有分支射线均保留在该端点内）总能强制的最大宽度，则W(d)=⌊2d/3⌋+1（对所有d≥4），且W(1)=1、W(2)=W(3)=2。
英文摘要：
Grid theorems provide a fundamental link between the structure of ends and the existence of grid-like subgraphs in infinite graphs and digraphs. For every fixed width $n$, let $k(n)$ denote the least positive integer such that every digraph with an end of in-degree at least $k(n)$ contains a subdivision of the directed hexagonal grid of width $n$. Hamann and Heuer asked the exact vaule of $k(n)$. We solve this problem by proving that $$ k(n)=\left\lfloor\frac{3n}{2}\right\rfloor-1 \qquad\text{for every }n\geq4, $$ while $k(1)=1$, $k(2)=2$, and $k(3)=4$. For the upper bound, we establish a finite vacancy theorem for labelled token slides, convert it into a closed directed schedule on an auxiliary ray digraph, and lift the schedule through fresh clean linkages. For the matching lower bound, we use alternating orientations of products of a ray with a three-armed tree and prove a no-passing property for disjoint directed paths. We also determine the dual extremal parameter. Let $W(d)$ denote the largest width that is always forced by an end of in-degree $d$, with all branch rays remaining in that end, then $$W(d)=\left\lfloor\frac{2d}{3}\right\rfloor+1 \qquad\text{for every }d\geq4,$$ with $W(1)=1$ and $W(2)=W(3)=2$.
