Forcing Generalized Quasirandom Graphs Efficiently

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Authors

GRZESIK Andrzej KRÁĽ Daniel PIKHURKO Oleg

Year of publication 2023
Type Article in Proceedings
Conference European Conference on Combinatorics, Graph Theory and Applications
MU Faculty or unit

Faculty of Informatics

Citation
Web https://journals.muni.cz/eurocomb/article/view/35604
Doi http://dx.doi.org/10.5817/CZ.MUNI.EUROCOMB23-070
Keywords graph limits; quasirandomness; stochastic block model
Description We study generalized quasirandom graphs whose vertex set consists of q parts (of not necessarily the same sizes) with edges within each part and between each pair of parts distributed quasirandomly; such graphs correspond to the stochastic block model studied in statistics and network science. Lovász and Sós showed that the structure of such graphs is forced by homomorphism densities of graphs with at most (10q)^q+q vertices; subsequently, Lovász refined the argument to show that graphs with 4(2q+3)^8 vertices suffice. Our results imply that the structure of generalized quasirandom graphs with q>=2 parts is forced by homomorphism densities of graphs with at most 4q^2-q vertices, and, if vertices in distinct parts have distinct degrees, then 2q+1 vertices suffice. The latter improves the bound of 8q-4 due to Spencer.
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