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arXiv · 1807.06947

Linearity of Saturation for Berge Hypergraphs

Sean English·Dániel Gerbner·Abhishek Methuku·Michael Tait
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Abstract

For a graph $F$, we say a hypergraph $H$ is Berge-$F$ if it can be obtained from $F$ be replacing each edge of $F$ with a hyperedge containing it. We say a hypergraph is Berge-$F$-saturated if it does not contain a Berge-$F$, but adding any hyperedge creates a copy of Berge-$F$. The $k$-uniform saturation number of Berge-$F$, $\mathrm{sat}_k(n,\text{Berge-}F)$ is the fewest number of edges in a Berge-$F$-saturated $k$-uniform hypergraph on $n$ vertices. We show that $\mathrm{sat}_k(n,\text{Berge-}F) = O(n)$ for all graphs $F$ and uniformities $3\leq k\leq 5$, partially answering a conjecture of English, Gordon, Graber, Methuku, and Sullivan. We also extend this conjecture to Berge copies of hypergraphs.

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Sean English, Dániel Gerbner, Abhishek Methuku, Michael Tait. 2018-07-18. Linearity of Saturation for Berge Hypergraphs. https://arxiv.org/abs/1807.06947

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