Asymptotic Structure of Graphs with the Minimum Number of Triangles

TL;DR

This paper investigates the asymptotic structure of graphs that minimize the number of triangles for given size and order, using flag algebra methods to characterize extremal configurations.

Contribution

It provides a detailed characterization of extremal graphs minimizing triangles, advancing understanding of their asymptotic structure through flag algebra analysis.

Findings

Identifies the asymptotic structure of extremal graphs with minimal triangles.

Characterizes the set of flag algebra homomorphisms that achieve minimal triangle density.

Provides a framework for analyzing triangle minimization in large graphs.

Abstract

We consider the problem of minimizing the number of triangles in a graph of given order and size and describe the asymptotic structure of extremal graphs. This is achieved by characterizing the set of flag algebra homomorphisms that minimize the triangle density.