Tree densities in sparse graph classes

TL;DR

This paper determines the maximum number of copies of a fixed forest in various sparse graph classes, revealing a unified approach based on stable set sizes in subforests induced by low-degree vertices.

Contribution

It provides a general framework for counting forest copies in sparse graphs using a single lemma related to excluded subgraphs.

Findings

Maximum copies of a fixed forest scale as (n^{\u03b1_d(T)})

Results apply to classes like k-degenerate, K_{s,t}-minor free, and k-planar graphs

A unified lemma underpins all the results

Abstract

What is the maximum number of copies of a fixed forest $T$ in an $n$-vertex graph in a graph class $\mathcal{G}$ as $n\to \infty$? We answer this question for a variety of sparse graph classes $\mathcal{G}$. In particular, we show that the answer is $\Theta(n^{\alpha_d(T)})$ where $\alpha_d(T)$ is the size of the largest stable set in the subforest of $T$ induced by the vertices of degree at most $d$, for some integer $d$ that depends on $\mathcal{G}$. For example, when $\mathcal{G}$ is the class of $k$-degenerate graphs then $d=k$; when $\mathcal{G}$ is the class of graphs containing no $K_{s,t}$-minor ($t\geq s$) then $d=s-1$; and when $\mathcal{G}$ is the class of $k$-planar graphs then $d=2$. All these results are in fact consequences of a single lemma in terms of a finite set of excluded subgraphs.