Towards characterizing locally common graphs
Abstract
A graph H is common if the number of monochromatic copies of H in a 2edgecoloring 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 Csóka, Hubai and Lovász [arXiv:1912.02926], where the graph is required to be the minimizer with respect to perturbations of the random 2edgecoloring. We give a complete analysis of the 12 initial terms in the Taylor series determining the number of monochromatic copies of H in such perturbations and classify graphs 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, and 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2020
 DOI:
 10.48550/arXiv.2011.02562
 arXiv:
 arXiv:2011.02562
 Bibcode:
 2020arXiv201102562H
 Keywords:

 Mathematics  Combinatorics