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On Homomorphism Graphs

Sebastian Brandt, Yi-Jun Chang, Jan Grebík, Christoph Grunau, Václav Rozhoň, Zoltán Vidnyánszky

TL;DR

This work develops a flexible framework of homomorphism graphs to study bounded-degree acyclic Borel graphs within descriptive combinatorics and the LOCAL model. By adapting Marks' determinacy technique to the Δ-regular tree setting, it shows that for every Δ>2 the class of Δ-regular acyclic Borel graphs with χ_B≤Δ is Σ^1_2-complete, signaling a fundamental barrier to simple Brooks-type characterizations in the Borel setting. The authors introduce Hom^{ac}(T_Δ, H) (and its edge-labeled variant) to transfer properties from a target graph H to acyclic Δ-regular Borel graphs, yielding notable results including hyperfinite constructions, complexity bounds, and nonexistence of Borel homomorphisms to finite graphs for certain parameter ranges. They also identify almost Δ-colorable graphs H_Δ with χ(H_Δ)=2Δ-2 and show how these parameters bound the feasibility of Borel homomorphisms, illustrating both maximal colorability and the limits of homomorphism-based characterizations in descriptive combinatorics.

Abstract

We introduce a new type of examples of bounded degree acyclic Borel graphs and study their combinatorial properties in the context of descriptive combinatorics, using a generalization of the determinacy method of Marks. The motivation for the construction comes from the adaptation of this method to the LOCAL model of distributed computing. Our approach unifies the previous results in the area, as well as produces new ones. In particular, we show that for $Δ>2$ it is impossible to give a simple characterization of acyclic $Δ$-regular Borel graphs with Borel chromatic number at most $Δ$: such graphs form a $\mathbfΣ^1_2$-complete set. This implies a strong failure of Brooks'-like theorems in the Borel context.

On Homomorphism Graphs

TL;DR

This work develops a flexible framework of homomorphism graphs to study bounded-degree acyclic Borel graphs within descriptive combinatorics and the LOCAL model. By adapting Marks' determinacy technique to the Δ-regular tree setting, it shows that for every Δ>2 the class of Δ-regular acyclic Borel graphs with χ_B≤Δ is Σ^1_2-complete, signaling a fundamental barrier to simple Brooks-type characterizations in the Borel setting. The authors introduce Hom^{ac}(T_Δ, H) (and its edge-labeled variant) to transfer properties from a target graph H to acyclic Δ-regular Borel graphs, yielding notable results including hyperfinite constructions, complexity bounds, and nonexistence of Borel homomorphisms to finite graphs for certain parameter ranges. They also identify almost Δ-colorable graphs H_Δ with χ(H_Δ)=2Δ-2 and show how these parameters bound the feasibility of Borel homomorphisms, illustrating both maximal colorability and the limits of homomorphism-based characterizations in descriptive combinatorics.

Abstract

We introduce a new type of examples of bounded degree acyclic Borel graphs and study their combinatorial properties in the context of descriptive combinatorics, using a generalization of the determinacy method of Marks. The motivation for the construction comes from the adaptation of this method to the LOCAL model of distributed computing. Our approach unifies the previous results in the area, as well as produces new ones. In particular, we show that for it is impossible to give a simple characterization of acyclic -regular Borel graphs with Borel chromatic number at most : such graphs form a -complete set. This implies a strong failure of Brooks'-like theorems in the Borel context.

Paper Structure

This paper contains 18 sections, 28 theorems, 32 equations, 2 figures.

Table of Contents

  1. Introduction
  2. a) Complexity result.
  3. b) Chromatic number and hyperfiniteness.
  4. c) Graph homomorphism.
  5. Roadmap.
  6. Preliminaries
  7. The homomorphism graph
  8. The $\Delta$-regular tree $T_\Delta$.
  9. Variations on Marks' technique
  10. Generalizations
  11. Edge labeled graphs.
  12. Graph homomorphism.
  13. Applications
  14. Complexity of the coloring problem
  15. Hyperfiniteness
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\mathcal{H}$ be a locally countable Borel graph. Then we have In particular, $\chi_{\Game \mathbf{\Delta}^1_1}(\mathcal{H})>\Delta$ holds if the Ramsey measurable (if $V(\mathcal{H})=[\mathbb{N}]^\mathbb{N}$), Baire measurable, or $\mu$ measurable chromatic number of $\mathcal{H}$ is $>\Delta$.

Figures (2)

  • Figure 1: The game $\mathbb{G}(x,i)$
  • Figure 2: The maximal graph that is almost $\Delta$-colorable for $\Delta=3$.

Theorems & Definitions (65)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4: conleyhyp
  • Theorem 1.5
  • Remark 1.6
  • Theorem 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Proposition 2.4
  • ...and 55 more