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A quantitative Lovász criterion for Property B

Published online by Cambridge University Press:  07 August 2020

Asaf Ferber
Affiliation:
Department of Mathematics, University of California, Irvine
Asaf Shapira
Affiliation:
School of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel
Corresponding
E-mail address:

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

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Paper
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.

References

Alon, N. and Spencer, J. (1992) The Probabilistic Method. Wiley.Google Scholar
Bernstein, F. (1908) Zur Theorie der trigonometrische Reihen. Leipz. Ber. 60 325328.Google Scholar
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Cherkashin, D. and Kozik, J. (2015) A note on random greedy coloring of uniform hypergraphs. Random Struct. Algorithms 47 407413.CrossRefGoogle Scholar
Lovász, L. (1979) Combinatorial Problems and Exercises. North-Holland.Google Scholar
Miller, E. W. (1937) On a property of families of sets. Comput. Rend. Varsovie 30 3138.Google Scholar
Pluhar, A. (2009) Greedy colorings of uniform hypergraphs. Random Struct. Algorithms 35 216221.CrossRefGoogle Scholar
Seymour, P. (1974) On the two-colouring of hypergraphs. Quart. J. Math. Oxford 25 303312.CrossRefGoogle Scholar

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