Densities of 3-vertex graphs

TL;DR

This paper characterizes the possible distributions of 3-vertex subgraph densities in large graphs, extending previous work by fully describing the set of all such densities across all pairs of parameters.

Contribution

It provides a complete description of the set of 3-vertex graph densities in large graphs for all pairs of density parameters, beyond the previously studied (d_0,d_3) plane.

Findings

Complete characterization of the set of 3-vertex graph densities.

Extension of previous results to all pairs of density parameters.

Provides a comprehensive understanding of subgraph density distributions.

Abstract

Let d_i(G) be the density of the 3-vertex i-edge graph in a graph G, i.e., the probability that three random vertices induce a subgraph with i edges. Let S be the set of all quadruples (d_0,d_1,d_2,d_3) that are arbitrary close to 3-vertex graph densities in arbitrary large graphs. Huang, Linial, Naves, Peled and Sudakov have recently determined the projection of the set S to the (d_0,d_3) plane. We determine the projection of the set S to all the remaining planes.