The Graph Density Domination Exponent

TL;DR

This paper investigates the density domination exponent in graph theory, establishing bounds and tools to relate homomorphism densities between different graphs, unifying several classical conjectures and theorems.

Contribution

It introduces new methods for estimating the density domination exponent and extends existing results to broader classes of graphs.

Findings

Established bounds for the density domination exponent.

Unified various classical results under a common framework.

Extended previous results to new graph regimes.

Abstract

For graphs $G$ and $H$, what relations can be determined between $t(G,W)$ and $t(H,W)$ for a general graph $W$? We study this problem through the framework of the density domination exponent, which is defined to be the smallest constant $c$ such that $t(G,W)\ge t(H,W)^c$ for every graph $W$. This broad generalization encompasses the Sidorenko conjecture, the Erd\H{o}s-Simonovits Theorem on paths, and a variety of other statements relating graph homomorphism densities. We introduce some general tools for estimating the density domination exponent, and extend previous results to new graph regimes.