Homomorphism counts in robustly sparse graphs

TL;DR

This paper establishes a unified framework for estimating the maximum number of homomorphisms, subgraphs, and induced subgraphs of a fixed graph in various sparse graph classes, solving longstanding open problems.

Contribution

It provides a general theorem that bounds homomorphism counts in robustly sparse graphs, extending extremal graph theory to new classes like nowhere dense graphs.

Findings

Determines maximum homomorphism counts in hereditary classes with bounded expansion.

Establishes asymptotic densities for homomorphisms and subgraphs in nowhere dense classes.

Applies to classes of graphs with geometric and topological properties.

Abstract

For a fixed graph $H$ and for arbitrarily large host graphs $G$, the number of homomorphisms from $H$ to $G$ and the number of subgraphs isomorphic to $H$ contained in $G$ have been extensively studied in extremal graph theory and graph limits theory when the host graphs are allowed to be dense. This paper addresses the case when the host graphs are robustly sparse and proves a general theorem that solves a number of open questions proposed since 1990s and strengthens a number of results in the literature. We prove that for any graph $H$ and any set ${\mathcal H}$ of homomorphisms from $H$ to members of a hereditary class ${\mathcal G}$ of graphs, if ${\mathcal H}$ satisfies a natural and mild condition, and contracting disjoint subgraphs of radius $O(\lvert V(H) \rvert)$ in members of ${\mathcal G}$ cannot create a graph with large edge-density, then an obvious lower bound for the size of ${\mathcal H}$ gives a good estimation for the size of ${\mathcal H}$. This result determines the maximum number of $H$-homomorphisms, the maximum number of $H$-subgraphs, and the maximum number $H$-induced subgraphs in graphs in any hereditary class with bounded expansion up to a constant factor; it also determines the exact value of the asymptotic logarithmic density for $H$-homomorphisms, $H$-subgraphs and $H$-induced subgraphs in graphs in any hereditary nowhere dense class. Hereditary classes with bounded expansion include (topological) minor-closed families and many classes of graphs with certain geometric properties; nowhere dense classes are the most general sparse classes in sparsity theory. Our machinery also allows us to determine the maximum number of $H$-subgraphs in the class of all $d$-degenerate graphs with any fixed $d$.