Towards characterizing locally common graphs

TL;DR

This paper investigates the classification of graphs based on their local extremal properties in edge colorings, providing a detailed analysis of initial terms in a Taylor series to distinguish between different classes of graphs.

Contribution

It offers a complete analysis of the first 12 terms in the Taylor series for locally perturbing random colorings, classifying graphs into three categories and deriving new conditions for commonality.

Findings

Classifies graphs into three categories based on local commonality.

Provides necessary conditions for a graph to be common.

Provides sufficient conditions for a graph to be not common.

Abstract

A graph H is common if the number of monochromatic copies of 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 Cs\'oka, Hubai and Lov\'asz [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 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.