Tree densities in sparse graph classes

Abstract

What is the maximum number of copies of a fixed forest in an -vertex graph in a graph class G as → ∞? We answer this question for a variety of sparse graph classes G. In particular, we show that the answer is Θ( ( ) ) where ( ) is the size of the largest stable set in the subforest of induced by the vertices of degree at most , for some integer that depends on G. For example, when G is the class of -degenerate graphs then = ; when G is the class of graphs containing no , -minor ( ) then = − 1; and when G is the class of -planar graphs then = 2. All these results are in fact consequences of a single lemma in terms of a finite set of excluded subgraphs.

The maximum number of paths of length three in a planar graph

“…It is well‐known that $$f(n,{P}_{2})=3n-6$$, and it follows from Theorem 3 that $$f(n,{P}_{3})={n}^{2}+3n-16$$ for $$n\ge 4$$. The order of magnitude of $$f(n,H)$$ when $$H$$ is a fixed tree was determined in [14] and for general $$H$$ (and in arbitrary surfaces) by Huynh et al [22] (see also [21] for results in general sparse settings). In particular, for a path on $$k$$ vertices, we have $$f(n,{P}_{k})={\rm{\Theta }}({n}^{\lfloor \frac{k-1}{2}\rfloor +1})$$.…”

Shallow Minors, Graph Products and Beyond Planar Graphs

“…Huynh and Wood [50] introduced the following variant of shortcut systems. A set P of paths in a graph G is a (k, d) * -shortcut system if:…”

“…Huynh and Wood [50] introduced the following generalisation of low-degree squares of graphs: for a graph G and integer d 1, let G (d) be the graph obtained from G by adding a clique on N G (v) for each vertex v ∈ V (G) with deg G (v) d. Huynh and Wood [50] observed that G (d) = G P where P is some (2, d) ⋆ -shortcut system.…”

“…For a bipartite graph H with bipartition (A, B), define the half-square H Dujmović et al [22] showed that every (g, d)-map graph G is a subgraph of G P 0 for some graph G 0 with Euler genus at most g and some (2, 1 2 d(d − 3))-shortcut system P. Note that Huynh and Wood [50] observed that P is a (2, d) * -shortcut system. By Lemma 6, every (g, d)-map graph G is a 1-shallow topological minor of…”

Bounding the number of odd paths in planar graphs via convex optimization

“…Following this result, Alon and Caro [1] determined N n K ( , ) m 2,  exactly for all m, where K m 2, is the complete 2-by-m bipartite graph. In a series of works [3,8,11,16], which culminated with a recent paper of Huynh, Joret and Wood [10], the asymptotic value of N n H ( , )…”