WEAKLY SATURATED HYPERGRAPHS AND A CONJECTURE OF TUZA

Asaf Shapira; Mykhaylo Tyomkyn

doi:10.1090/proc/16197

WEAKLY SATURATED HYPERGRAPHS AND A CONJECTURE OF TUZA

Asaf Shapira, Mykhaylo Tyomkyn

School of Mathematical Sciences

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

Abstract

Given a fixed hypergraph H, let wsat(n,H) denote the smallest number of edges in an n-vertex hypergraph G, with the property that one can sequentially add the edges missing from G, so that whenever an edge is added, a new copy of H is created. The study of wsat(n, H) was introduced by Bollobás in 1968, and turned out to be one of the most influential topics in extremal combinatorics. While for most H very little is known regarding wsat(n, H), Alon proved in 1985 that for every graph H there is a limiting constant C_H so that wsat(n, H) = (C_H + o(1))n. Tuza conjectured in 1992 that Alon’s theorem can be (appropriately) extended to arbitrary r-uniform hypergraphs. In this paper we prove this conjecture.

Original language	English
Pages (from-to)	2795-2805
Number of pages	11
Journal	Proceedings of the American Mathematical Society
Volume	151
Issue number	7
DOIs	https://doi.org/10.1090/proc/16197
State	Published - 1 Jul 2023

All Science Journal Classification (ASJC) codes

General Mathematics
Applied Mathematics

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title = "WEAKLY SATURATED HYPERGRAPHS AND A CONJECTURE OF TUZA",

abstract = "Given a fixed hypergraph H, let wsat(n,H) denote the smallest number of edges in an n-vertex hypergraph G, with the property that one can sequentially add the edges missing from G, so that whenever an edge is added, a new copy of H is created. The study of wsat(n, H) was introduced by Bollob{\'a}s in 1968, and turned out to be one of the most influential topics in extremal combinatorics. While for most H very little is known regarding wsat(n, H), Alon proved in 1985 that for every graph H there is a limiting constant CH so that wsat(n, H) = (CH + o(1))n. Tuza conjectured in 1992 that Alon{\textquoteright}s theorem can be (appropriately) extended to arbitrary r-uniform hypergraphs. In this paper we prove this conjecture.",

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WEAKLY SATURATED HYPERGRAPHS AND A CONJECTURE OF TUZA

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