Hamming图 H(d,3)=K_3^{box d} 的交叉数与单位距离交叉数及其在多个坐标域上的实现
The crossing number and the unit-distance crossing number of the Hamming graphs H(d,3)=K_3^{box d}, and their realizations over many coordinate fields
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中文总结 AI 辅助
本文研究Hamming图H(d,3)的平面单位距离绘制，证明其普通交叉数为Theta(n^2)（常数7/6），给出单位距离交叉数的上下界，并揭示其可构造性与超越实现的几何统一。
中文摘要 AI 辅助
Hamming图 H(d,3)=K_3^{box d}（具有 n=3^d 个顶点）是三元码的单符号错误图，但我们对其提出一个几何问题：能否在平面上绘制该图，使得每条边长度恰好为一个单位，且任意两个非相邻顶点之间的距离不等于一个单位？对于每个 d，答案都是肯定的，这同时将两个问题合二为一。该图位于何处？作为单位三角形的闵可夫斯基和，每个 H(d,3) 都具有隐藏的灵活性，可将其坐标沿可构造性阶梯（圆规、折纸等）提升，因此同一个图可在多个数域上同时实现，既可在平面中也可在 R^3 中（edim(H(d,q))=q-1）。然而，几乎所有忠实实现都是超越的：伽罗瓦图景只是广阔超越连续统的零测度影子，这与单位距离识别的 exists-R 难度相匹配。单位绘制必须有多拥挤？我们将普通交叉数与单位距离交叉数（Schaefer 的综述中未包含）区分开来，一个递归式 H(d,3)=H(d-1,3) box K_3 同时控制两者。我们证明 cr(H)=Theta(n^2)，并给出精确的一页常数 7/6；以及通过集中不等式得到 Omega(n^2) <= udcr(H) <= O(n^2 log n)，外加一个闭式上界 (3/2)n^2(L^2-L+1)，其中 L=log_3 n。我们找到的最小交叉绘制是可构造的（折纸）：域和交叉是同一几何的两种解读。所有结论均经计算验证；下界 udcr=Omega(n^2 log n) 是核心开放问题。
英文摘要
The Hamming graph H(d,3)=K_3^{box d} (n=3^d vertices) is the graph of single-symbol errors of ternary codes, yet we ask a geometric question of it: can it be drawn in the plane with every edge exactly one unit long and no two non-adjacent vertices a unit apart? It can, for every d, joining two problems into one. Where does it live? As a Minkowski sum of unit triangles, each H(d,3) has a hidden flexibility carrying its coordinates up the constructibility ladder (compass, origami, and beyond), so one graph is realizable over many number fields at once, in the plane and in R^3 (edim(H(d,q))=q-1). Yet almost every faithful realization is transcendental: the Galois picture is a measure-zero shadow of a vast transcendental continuum, matching the exists-R hardness of unit-distance recognition. How crowded must a unit drawing be? We separate the ordinary crossing number from a unit-distance crossing number (absent from Schaefer's survey), and one recursion H(d,3)=H(d-1,3) box K_3 controls both. We prove cr(H)=Theta(n^2), with sharp one-page constant 7/6; and Omega(n^2) <= udcr(H) <= O(n^2 log n) by concentration, plus a closed-form majorant (3/2)n^2(L^2-L+1), L=log_3 n. The least-crossing drawing we find is constructible (origami): field and crossings are two readings of one geometry. All claims are verified computationally; the lower bound udcr=Omega(n^2 log n) is the central open problem.
发表机构
- Universidade do Estado do Rio de Janeiro (UERJ)（里约热内卢州立大学）
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