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Fractional Isomorphism of Graphons


We work out the theory of fractional isomorphism of graphons as a generalization to the classical theory of fractional isomorphism of finite graphs. The generalization is given in terms of homomorphism densities of finite trees and it is characterized in terms of distributions on iterated degree measures, Markov operators, weak isomorphism of a conditional expectation with respect to invariant sub-σ-algebras and isomorphism of certain quotients of given graphons.

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The authors are grateful to Jan Hladky for useful discussions and help, and to anonymous referees for many useful suggestions and comments that helped to improve the presentation of the paper. The first author also thanks Jan Bydžovský, Jan Hladký and Oleg Pikhurko for help with the current version of the introduction.

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Correspondence to Jan Grebík.

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Grebík was supported by the Czech Science Foundation, grant number GJ16-07822Y, by the grant GAUK 900119 of Charles University and by Leverhulme Research Project Grant RPG-2018-424. Part of the work was done while Grebík was affiliated with Institute of Computer Science of the Czech Academy of Sciences, with institutional support RVO:67985807.

Rocha was supported by the Czech Science Foundation, grant number GJ16-07822Y and GA19-08740S. With institutional support RVO:67985807.

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Grebík, J., Rocha, I. Fractional Isomorphism of Graphons. Combinatorica 42, 365–404 (2022).

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Mathematics Subject Classification (2010)

  • 05C80
  • 05C50
  • 05C80
  • 05C60