Complexity of Borel Chromatic Number for Bounded Degree Acyclic Graphs

Core Concepts

The Borel chromatic number of bounded degree acyclic Borel graphs does not admit a simple characterization and is Σ1
2-complete.

Abstract

The paper introduces a new type of examples of bounded degree acyclic Borel graphs, called homomorphism graphs, and studies their combinatorial properties in the context of descriptive combinatorics. The main result shows that for ∆> 2, the family of ∆-regular acyclic Borel graphs with Borel chromatic number at most ∆forms a Σ1
2-complete set. This implies a strong failure of Brooks'-like theorems in the Borel context.
The key idea is to associate an acyclic Borel graph Homac(T∆, H) to a given Borel graph H, where the vertex set consists of pairs (x, h) with x a vertex in H and h a homomorphism from the ∆-regular infinite rooted tree T∆to H that maps the root to x. The authors show that the Borel chromatic number of Homac(T∆, H) is controlled by the weakly provably ∆1
2-chromatic number of H.
Using this construction, the authors establish several applications:

A complexity result showing that the family of ∆-regular acyclic Borel graphs with Borel chromatic number at most ∆has no simple characterization and is Σ1
2-complete.
A new, streamlined proof of the existence of hyperfinite ∆-regular acyclic Borel graphs with Borel chromatic number ∆+ 1.
The construction of ∆-regular acyclic Borel graphs that do not admit Borel homomorphisms to finite graphs of chromatic number at most ∆.

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Key Insights Distilled From

On Homomorphism Graphs

by Seba... at arxiv.org 05-01-2024

https://arxiv.org/pdf/2111.03683.pdf

Deeper Inquiries

Can the homomorphism graph construction be extended to other classes of graphs beyond bounded degree acyclic Borel graphs to obtain similar complexity results

The homomorphism graph construction can potentially be extended to other classes of graphs beyond bounded degree acyclic Borel graphs to obtain similar complexity results. By adapting the construction to different types of graphs with specific properties, it may be possible to explore the combinatorial properties and chromatic numbers of those graphs in the context of descriptive combinatorics. For example, one could consider extending the construction to graphs with specific connectivity properties, edge labelings, or other structural characteristics to study their chromatic numbers and combinatorial properties. This extension could lead to new insights into the complexity of coloring problems for various classes of graphs.

Are there other applications of the homomorphism graph technique in the context of descriptive combinatorics or distributed computing

The homomorphism graph technique can have several other applications in the context of descriptive combinatorics and distributed computing. Here are a few potential applications:

Ramsey Theory: The homomorphism graph technique can be used to study Ramsey properties of graphs. By analyzing the existence of homomorphisms between graphs, one can investigate Ramsey properties and related combinatorial problems in graph theory.

Algorithm Design: The construction of homomorphism graphs can provide insights into algorithm design for distributed computing. By understanding the relationships between graphs and their homomorphisms, one can develop efficient algorithms for tasks such as graph coloring, connectivity, and optimization in distributed systems.

Network Analysis: Homomorphism graphs can be used to analyze and model complex networks in various domains, such as social networks, communication networks, and biological networks. By studying the homomorphisms between different network structures, one can gain a deeper understanding of network properties and behaviors.

Complexity Theory: The study of homomorphism graphs can contribute to complexity theory by providing insights into the computational complexity of graph problems. By analyzing the existence of homomorphisms between graphs with specific properties, one can investigate the complexity of decision problems and optimization tasks on graphs.

What are the implications of the failure of Brooks'-like theorems in the Borel context for practical problems in areas like distributed algorithms or network design

The failure of Brooks'-like theorems in the Borel context has significant implications for practical problems in areas like distributed algorithms and network design. Some implications include:

Algorithmic Challenges: The inability to characterize acyclic Borel graphs with Borel chromatic numbers in certain cases poses algorithmic challenges for distributed algorithms. It indicates the complexity of coloring problems in distributed systems and highlights the need for sophisticated algorithms to handle such complexities.

Network Optimization: In network design and optimization, the failure of Brooks'-like theorems suggests that simple characterizations of graph properties may not hold in the Borel context. This implies that network designers need to consider more complex models and algorithms to optimize network structures and properties.

Distributed Computing: In the context of distributed computing, the failure of Brooks'-like theorems can impact the design and analysis of distributed algorithms. It indicates that traditional theorems and results may not directly apply in the Borel context, requiring researchers to develop new approaches and techniques for solving distributed computing problems.

Graph Coloring: The results on acyclic Borel graphs and their chromatic numbers have implications for graph coloring problems in distributed systems. Understanding the complexity of coloring problems in the Borel context can help in designing efficient distributed algorithms for tasks that involve graph coloring and related combinatorial problems.