Tree densities in sparse graph classes

Abstract

AbstractWhat is the maximum number of copies of a fixed forestTin ann-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)$is the size of the largest stable set in the subforest ofTinduced by the vertices of degree at mostd, for some integerdthat depends on$\mathcal {G}$. For example, when$\mathcal {G}$is the class ofk-degenerate graphs then$d=k$; when$\mathcal {G}$is the class of graphs containing no$K_{s,t}$-minor ($t\geqslant s$) then$d=s-1$; and when$\mathcal {G}$is the class ofk-planar graphs then$d=2$. All these results are in fact consequences of a single lemma in terms of a finite set of excluded subgraphs.