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Strongly sublinear separators and bounded asymptotic dimension for sphere intersection graphs

James Davies, Agelos Georgakopoulos, Meike Hatzel, Rose McCarty

TL;DR

This work studies sphere intersection graphs in $\mathbb{R}^{d}$, introducing the concept of sphere dimension and linking it to circle graphs. It proves two main results: first, for every $t,d\ge 2$, the class $\mathcal{C}^{d}_{t}$ (dimension-bounded graphs with no $K_{t,t}$ subgraph) has strongly sublinear balanced separators of size $\widetilde{O}_{d}( t^{10}n^{1-\frac{1}{2d+8}} )$, and second, the full class $\mathcal{C}^{d}$ has asymptotic dimension at most $2d+2$. The asymptotic-dimension result is extended to a broader class $\mathcal{C}^{d}_{\alpha}$ of convex-set intersections via a layerable, quasi-isometric framework, building on work by Dvořák–Norin and Bonamy et al. These findings connect geometric representations to algorithmic partitioning and coloring properties, with implications for sparse-partition schemes and coarse embeddings in high dimensions.

Abstract

In this paper, we consider the class $\mathcal{C}^d$ of sphere intersection graphs in $\mathbb{R}^d$ for $d \geq 2$. We show that for each integer $t$, the class of all graphs in $\mathcal{C}^d$ that exclude $K_{t,t}$ as a subgraph has strongly sublinear separators. We also prove that $\mathcal{C}^d$ has asymptotic dimension at most $2d+2$.

Strongly sublinear separators and bounded asymptotic dimension for sphere intersection graphs

TL;DR

This work studies sphere intersection graphs in , introducing the concept of sphere dimension and linking it to circle graphs. It proves two main results: first, for every , the class (dimension-bounded graphs with no subgraph) has strongly sublinear balanced separators of size , and second, the full class has asymptotic dimension at most . The asymptotic-dimension result is extended to a broader class of convex-set intersections via a layerable, quasi-isometric framework, building on work by Dvořák–Norin and Bonamy et al. These findings connect geometric representations to algorithmic partitioning and coloring properties, with implications for sparse-partition schemes and coarse embeddings in high dimensions.

Abstract

In this paper, we consider the class of sphere intersection graphs in for . We show that for each integer , the class of all graphs in that exclude as a subgraph has strongly sublinear separators. We also prove that has asymptotic dimension at most .

Paper Structure

This paper contains 6 sections, 15 theorems, 3 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sphere graphs and sphere dimension
  4. Strongly sublinear separators
  5. Asymptotic dimension
  6. Final remarks and open problems

Key Result

Theorem 0

For any integers $t,d \geq 2$, the class $\mathcal{C}^{d}_t$ has strongly sublinear separators. More precisely, every $n$-vertex graph $G \in \mathcal{C}^{d}_t$ has a balanced separator of size $\widetilde{\mathcal{O}}_d( t^{10}n^{1-\frac{1}{2d+8}} )$.

Figures (2)

  • Figure 1: This illustrates the case that many vertices of $\mathcal{N}$ lie in a single bag. The set $\mathcal{X}=\left\lbrace X_1,\dots, X_6 \right\rbrace$ is depicted together with the corresponding leaves (note that $X_3$ is equal to its leaf). Using two paired bags, the green paths are of length at most $6r+1$ and pairwise disjoint.
  • Figure 2: This illustrates the case that many vertices of $\mathcal{N}$ lie in different bags. The outermost spheres are mapped to the innermost spheres such that the green paths (all of length at most $4r+2$) have to cross the spheres lying in between.

Theorems & Definitions (29)

  • Theorem 0
  • Theorem 1
  • Theorem 2: Plotkin, Rao, and Smith PRS
  • Lemma 3: 2022plyballscol
  • proof
  • proof
  • proof
  • Lemma 8
  • proof
  • Lemma 9
  • ...and 19 more