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Pré-Publication, Document De Travail Année : 2024

Holey graphs: very large Betti numbers are testable

Résumé

We show that the graph property of having a (very) large $k$-th Betti number $\beta_k$ for constant $k$ is testable with a constant number of queries in the dense graph model. More specifically, we consider a clique complex defined by an underlying graph and prove that for any $\varepsilon>0$, there exists $\delta(\varepsilon,k)>0$ such that testing whether $\beta_k \geq (1-\delta) d_k$ for $\delta \leq \delta(\varepsilon,k)$ reduces to tolerantly testing $(k+2)$-clique-freeness, which is known to be testable. This complements a result by Elek (2010) showing that Betti numbers are testable in the bounded-degree model. Our result combines the Euler characteristic, matroid theory and the graph removal lemma.
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Dates et versions

hal-04516209 , version 1 (22-03-2024)

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Dániel Szabó, Simon Apers. Holey graphs: very large Betti numbers are testable. 2024. ⟨hal-04516209⟩
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