关于连通性图码的Alon问题：对所有d≥4，f(d)=2^d
Alon's Question on Connectivity Graph-Codes: $f(d)=2^d$ for Every $d\geq 4$
- Princeton University（普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究肯定回答了Alon关于连通性图码的问题，证明对所有d≥4，f(d)=2^d，并构造无穷多带线性连通性图码的d-正则二分图。
AI中文摘要：
对于有限图H，连通性图码是满足：当A和B是𝒞的不同元素时，A△B是H的连通生成子图的族𝒞⊆2^{E(H)}。令m(H)表示该族的最大规模，f(d)表示使得无穷多个两两非同构的d-正则图H满足m(H)=q的最大整数q。将码字限制在与一个顶点关联的边，可得f(d)≤2^d。Alon证明了对所有足够大的d等式成立，并询问是否对所有d≥4均成立。我们肯定回答了该问题。更准确地说，对每个d≥4，我们构造了无穷多个有限单d-正则二分图，其带有维数为d的线性连通性图码。该构造始于一个向量标记的K_{d,d}副本；对于d≥7，所需标记来自GL_d(2)中不可约共轭类的概率计数，经简短穷尽程序验证的显式矩阵覆盖d=4,5,6；循环电压提升则生成所需的无穷族。
英文摘要：
For a finite graph $H$, a connectivity graph-code is a family $\mathcal C\subseteq 2^{E(H)}$ such that $A\triangle B$ is a connected spanning subgraph of $H$ whenever $A$ and $B$ are distinct members of $\mathcal C$. Let $m(H)$ denote the maximum size of such a family, and let $f(d)$ be the largest integer $q$ for which $m(H)=q$ for infinitely many pairwise nonisomorphic $d$-regular graphs $H$. Restricting codewords to the edges incident with a vertex gives $f(d)\leq 2^d$. Alon proved equality for all sufficiently large $d$ and asked whether it holds for every $d\geq 4$. We answer this question affirmatively. More precisely, for every $d\geq 4$ we construct infinitely many finite simple $d$-regular bipartite graphs carrying a linear connectivity graph-code of dimension $d$. The construction begins with a vector-labelled copy of $K_{d,d}$. For $d\geq 7$, the required labelling follows from a probabilistic count over an irreducible conjugacy class in $\mathrm{GL}_d(2)$; explicit matrices, verified by a short exact exhaustive program, cover $d=4,5,6$. Cyclic voltage lifts then produce the required infinite families.
