发表机构
University of Delaware; University of California, San Diego（特拉华大学; 加州大学圣地亚哥分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文研究强正则图的VC维，刻画了VC维为2的足够大的强正则图，确定了拉丁方图及阶数至多28的强正则图的VC维，并证明最小特征值为给定整数的强正则图VC维有界。
AI 中文摘要
一个图$G$是$n$-存在封闭的或$n$-e.c.的，如果对于所有满足$|S|=n$的子集$S\subseteq V(G)$以及所有划分$S=A\sqcup B$，存在一个顶点在$V(G)\sm S$中，与$A$中的所有顶点相邻且与$B$中的任何顶点不相邻。我们研究具有$v$个顶点的$n$-e.c.图的最小边数$m(v,n)$，并证明$m(v,2)=3v+O(1)$，而对于固定的$n\ge 3$，$m(v,n)=\Theta(v\log v)$。后一个结果利用了与二元覆盖阵列的联系。一个相关参数是$G$的VC维，定义为被$G$中顶点的邻域打碎的最大顶点子集的大小。我们开始了对强正则图（SRGs）的VC维的系统研究。我们刻画了足够大的VC维为2的SRGs。此外，我们确定了足够大的拉丁方图以及所有阶数至多28的SRGs的VC维，并证明了具有给定整数作为最小特征值的SRGs具有有界VC维。
英文摘要
A graph $G$ is $n$-existentially closed or $n$-e.c. if, for all subsets $S\subseteq V(G)$ with $|S|=n$ and for all partitions $S=A\sqcup B$, there exists a vertex in $V(G)\sm S$ adjacent to all vertices in $A$ and no vertices in $B$. We study the minimum number of edges $m(v,n)$ of a $v$-vertex $n$-e.c. graph, and show that $m(v,2)=3v+O(1)$ while $m(v,n)=Θ(v\log v)$ for fixed $n\ge 3$. The latter result uses a connection to binary covering arrays. A related parameter is the VC-dimension of $G$, defined as the size of the largest subset of vertices shattered by the neighborhoods of vertices in $G$. We initiate systematic study of the VC-dimensions of strongly regular graphs (SRGs). We characterize the sufficiently large SRGs with VC-dimension 2. Furthermore, we determine the VC-dimension of sufficiently large Latin square graphs and of all SRGs of order at most 28, and we show that the SRGs with a given integer as smallest eigenvalue have bounded VC-dimension.
Comments28 pages, 6 figures
