Fractionally Balanced Hypergraphs and Rainbow KKM Theorems

Ron Aharoni, Eli Berger, Joseph Briggs, Erel Segal-Halevi, Shira Zerbib

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1 Scopus citations

Abstract

A d-partite hypergraph is called fractionally balanced if there exists a non-negative, not identically zero, function on its edge set that has constant degrees in each vertex side. Using a topological version of Hall’s theorem we prove lower bounds on the matching number of such hypergraphs. These bounds yield rainbow versions of the KKM theorem for products of simplices, which in turn are used to obtain some results on multiple-cake division, and on rainbow matchings in families of d-intervals.

Original languageEnglish
Pages (from-to)913-951
Number of pages39
JournalCombinatorica
Volume42
StatePublished - Dec 2022
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2022, János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg.

Funding

R. Aharoni was Supported by the Israel Science Foundation (ISF) grant no. 2023464 and the Discount Bank Chair at the Technion. This paper is part of a project that has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 823748. This work was supported by the Russian Federation Government in the framework of MegaGrant no. 075-15-2019-1926 when Ron Aharoni worked on Sections 6 through 8 of the paper. Erel Segal-Halevi was supported by the Israel Science Foundation (grant no. 712/20). R. Aharoni, E. Berger and S. Zerbib are supported by BSF grant no. 2016077. Acknowledgements Shira Zerbib was supported by NSF grant DMS-1953929.

FundersFunder number
Russian Federation Government075-15-2019-1926
National Science FoundationDMS-1953929
Bloom's Syndrome Foundation2016077
Horizon 2020 Framework Programme823748
Israel Science Foundation712/20, 2023464
Technion-Israel Institute of Technology

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