Resilient Hypergraphs with Fixed Matching Number

Combinatorica volume 38, pages1079–1094(2018)Cite this article

Abstract

Let H be a hypergraph of rank k, that is, |H| ≦ k for all H ∈ H. Let ν(H) denote the matching number, the maximum number of pairwise disjoint edges in H. For a vertex x let H(x̄) be the hypergraph consisting of the edges H ∈ H with x ∉ H. If ν(H(x̄)) = ν(H) for all vertices, H is called resilient. The main result is the complete determination of the maximum number of 2-element sets in a resilient hypergraph with matching number s. For k=3 it is \(\left( {\begin{array}{*{20}c} {2s + 1} \\ 2 \\ \end{array} } \right)\) while for k ≧ 4 the formula is \(k \cdot \left( {\begin{array}{*{20}c} {2s + 1} \\ 2 \\ \end{array} } \right)\). The results are used to obtain a stability theorem for k-uniform hypergraphs with given matching number.

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  1. Alfréd Rényi Institute, Budapest, Hungary

    Peter Frankl

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  1. Peter Frankl
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Correspondence to Peter Frankl.

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Frankl, P. Resilient Hypergraphs with Fixed Matching Number. Combinatorica 38, 1079–1094 (2018). https://doi.org/10.1007/s00493-016-3579-3

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Mathematics Subject Classification (2000)

  • 05D05
  • 05C65