The Graph Density Domination Exponent

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 $t(G,W)\ge t(H,W)<sup>c$ 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.