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arXiv · 1903.04968

A quantitative Lovász criterion for Property B

Asaf Ferber·Asaf Shapira
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Abstract

A well known observation of Lovász is that if a hypergraph is not $2$-colorable, then at least one pair of its edges intersect at a single vertex. %This very simple criterion turned out to be extremly useful . In this short paper we consider the quantitative version of Lovász's criterion. That is, we ask how many pairs of edges intersecting at a single vertex, should belong to a non $2$-colorable $n$-uniform hypergraph? Our main result is an {\em exact} answer to this question, which further characterizes all the extremal hypergraphs. The proof combines Bollobás's two families theorem with Pluhar's randomized coloring algorithm.

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Asaf Ferber, Asaf Shapira. 2019-03-12. A quantitative Lovász criterion for Property B. https://doi.org/10.1017/s0963548320000334

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