Toward characterizing locally common graphs

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Authors

HANCOCK Robert Arthur KRÁĽ Daniel KRNC Matjaz VOLEC Jan

Year of publication 2023
Type Article in Periodical
Magazine / Source RANDOM STRUCTURES & ALGORITHMS
MU Faculty or unit

Faculty of Informatics

Citation
Web https://onlinelibrary.wiley.com/doi/10.1002/rsa.21099
Doi http://dx.doi.org/10.1002/rsa.21099
Keywords common graphs; graph limits; Ramsey theory
Description A graph H$$ H $$ is common if the number of monochromatic copies of H$$ H $$ in a 2-edge-coloring of the complete graph is asymptotically minimized by the random coloring. The classification of common graphs is one of the most intriguing problems in extremal graph theory. We study the notion of weakly locally common graphs considered by Csoka, Hubai, and Lovasz [arXiv:1912.02926], where the graph is required to be the minimizer with respect to perturbations of the random 2-edge-coloring. We give a complete analysis of the 12 initial terms in the Taylor series determining the number of monochromatic copies of H$$ H $$ in such perturbations and classify graphs H$$ H $$ based on this analysis into three categories: Graphs of Class I are weakly locally common. Graphs of Class II are not weakly locally common. Graphs of Class III cannot be determined to be weakly locally common or not based on the initial 12 terms. As a corollary, we obtain new necessary conditions on a graph to be common and new sufficient conditions on a graph to be not common.
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