Locally common graphs

TL;DR

This paper investigates the concept of locally common graphs, showing that graphs containing K4 are not locally common but are weakly locally common, and that not all connected graphs share this property.

Contribution

It introduces and analyzes two notions of locally common graphs, sharpening existing results and exploring their limitations within graph limit theory.

Findings

Graphs containing K4 are not locally common.

All graphs containing K4 are weakly locally common.

Not all connected graphs are weakly locally common.

Abstract

Goodman proved that the sum of the number of triangles in a graph on $n$ nodes and its complement is at least $n^3/24$; in other words, this sum is minimized, asymptotically, by a random graph with edge density $1/2$. Erd\H{o}s conjectured that a similar inequality will hold for $K_4$ in place of $K_3$, but this was disproved by Thomason. But an analogous statement does hold for some other graphs, which are called {\it common graphs}. A characterization of common graphs seems, however, out of reach. Franek and R\"odl proved that $K_4$ is common in a weaker, local sense. Using the language of graph limits, we study two versions of locally common graphs. We sharpen a result of Jagger, \v{S}tov\'{\i}\v{c}ek and Thomason by showing that no graph containing $K_4$ can be locally common, but prove that all such graphs are weakly locally common. We also show that not all connected graphs are weakly locally common.