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A quantitative Lovász criterion for Property B

Part of: Graph theory

Published online by Cambridge University Press:  07 August 2020

Asaf Ferber  and
Asaf Shapira
Asaf Ferber*
Affiliation:
Department of Mathematics, University of California, Irvine
Asaf Shapira
Affiliation:
School of Mathematics, Tel Aviv University, Tel Aviv69978, Israel
*
*Corresponding author. Email: asaff@uci.edu
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Abstract

A well-known observation of Lovász is that if a hypergraph is not 2-colourable, then at least one pair of its edges intersect at a single vertex. 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-colourable n-uniform hypergraph. Our main result is an exact answer to this question, which further characterizes all the extremal hypergraphs. The proof combines Bollobás’s two families theorem with Pluhar’s randomized colouring algorithm.

MSC classification

Primary: 05C12: Distance in graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 6 , November 2020 , pp. 956 - 960
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

Supported in part by an NSF grant DMS-1954395.

‡

Supported in part by ISF Grant 1028/16 and ERC Starting Grant 633509.

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