A Topological Embedding of the Binary Tree into the Square Lattice

Abstract

We prove that for any finite tree $T$ with $n$ vertices and maximal degree $3$, there is a topological embedding of $T$ into the integer grid $Z^2$ which maps vertices to vertices and whose image meets at most $\frac{7}{3}n$ vertices. This recovers a weaker form of a result due to Valiant 10.5555/1963635.1963641 with stronger constants. We address question $7.7$ of arXiv:2112.05305, giving the first example of a pair of graphs $X,Y$ such that there is no regular map $X\to Y$ but the coarse wiring profile of $X$ into $Y$ grows linearly.