Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Topological Embedding of the Binary Tree into the Square Lattice

Samuel Kelly

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.

A Topological Embedding of the Binary Tree into the Square Lattice

Abstract

We prove that for any finite tree with vertices and maximal degree , there is a topological embedding of into the integer grid which maps vertices to vertices and whose image meets at most vertices. This recovers a weaker form of a result due to Valiant 10.5555/1963635.1963641 with stronger constants. We address question of arXiv:2112.05305, giving the first example of a pair of graphs such that there is no regular map but the coarse wiring profile of into grows linearly.
Paper Structure (2 sections, 5 theorems, 14 equations, 1 figure)

This paper contains 2 sections, 5 theorems, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The $1$-wiring profile $T_3 \to \mathbb{Z}^2$ is linear

Key Result

Theorem 1.0.3

Let $T_3$ denote the infinite $3$-regular tree, and let $\mathbb{Z}^2$ denote the 2-dimensional integer lattice. We have

Figures (1)

  • Figure 1: The image of $B_4$

Theorems & Definitions (11)

  • Definition 1.0.1: Wiring
  • Definition 1.0.2: Wiring Profile
  • Theorem 1.0.3
  • Corollary 1.0.3.1
  • proof
  • Lemma 2.0.1
  • proof
  • Lemma 2.0.2
  • proof
  • Lemma 2.0.3
  • ...and 1 more