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arXiv · 2609.10330

The VC-dimension of strongly regular graphs

Isabel Byrne·John Byrne·Sebastian M. Cioabă
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Abstract

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 $m(v,n)=\Theta(v\log v)$ 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.

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Isabel Byrne, John Byrne, Sebastian M. Cioabă. 2026-09-09. The VC-dimension of strongly regular graphs. https://arxiv.org/abs/2609.10330

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