David Wood, Tree densities of sparse graph classes

This talk considers the following question at the intersection of extremal and structural graph theory: What is the maximum number of copies of a fixed forest $T$ in an $n$-vertex graph in a graph class $\mathcal{G}$ as $n\to \mathrm{\infty }$? I will answer this question for a variety of sparse graph classes $\mathcal{G}$. In particular, we show that the answer is $\mathrm{\Theta }(n^{{\alpha }_d(T)})$ 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 ($t\ge s$) 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. This is joint work with Tony Huynh (arXiv:2009.12989).