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Borel Vizing's Theorem for Graphs of Subexponential Growth

Anton Bernshteyn, Abhishek Dhawan

TL;DR

This paper proves a Borel analogue of Vizing's theorem for graphs of subexponential growth: any Borel graph $G$ with finite maximum degree $\Delta$ and subexponential growth satisfies $\chi'_{\mathsf{B}}(G) \le \Delta + 1$. The authors derive a stronger finite-result—a deterministic $\mathsf{LOCAL}$ algorithm colors an $n$-vertex graph of subexponential growth with $\Delta+1$ colors in $O(\log^* n)$ rounds, with constants depending on the growth bound—by leveraging Christiansen's small augmenting subgraphs. They then transfer this finite-case outcome to the Borel setting via Bernshteyn's transfer principle, establishing the global bound for Borel graphs. The work combines descriptive set theoretic and distributed computing techniques, yielding a robust bridge between finite algorithmic guarantees and Borel graph colorings, and extends the scope of Vizing-type results to broader growth regimes. This advances the understanding of when combinatorial colorings admit Borel realizations and demonstrates the practical impact of distributed-algorithm tools in descriptive combinatorics.

Abstract

We show that every Borel graph $G$ of subexponential growth has a Borel proper edge-coloring with $Δ(G) + 1$ colors. We deduce this from a stronger result, namely that an $n$-vertex (finite) graph $G$ of subexponential growth can be properly edge-colored using $Δ(G) + 1$ colors by an $O(\log^\ast n)$-round deterministic distributed algorithm in the $\mathsf{LOCAL}$ model, where the implied constants in the $O(\cdot)$ notation are determined by a bound on the growth rate of $G$.

Borel Vizing's Theorem for Graphs of Subexponential Growth

TL;DR

This paper proves a Borel analogue of Vizing's theorem for graphs of subexponential growth: any Borel graph with finite maximum degree and subexponential growth satisfies . The authors derive a stronger finite-result—a deterministic algorithm colors an -vertex graph of subexponential growth with colors in rounds, with constants depending on the growth bound—by leveraging Christiansen's small augmenting subgraphs. They then transfer this finite-case outcome to the Borel setting via Bernshteyn's transfer principle, establishing the global bound for Borel graphs. The work combines descriptive set theoretic and distributed computing techniques, yielding a robust bridge between finite algorithmic guarantees and Borel graph colorings, and extends the scope of Vizing-type results to broader growth regimes. This advances the understanding of when combinatorial colorings admit Borel realizations and demonstrates the practical impact of distributed-algorithm tools in descriptive combinatorics.

Abstract

We show that every Borel graph of subexponential growth has a Borel proper edge-coloring with colors. We deduce this from a stronger result, namely that an -vertex (finite) graph of subexponential growth can be properly edge-colored using colors by an -round deterministic distributed algorithm in the model, where the implied constants in the notation are determined by a bound on the growth rate of .
Paper Structure (3 sections, 9 theorems, 7 equations)

This paper contains 3 sections, 9 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. From Theorem \ref{['theo:dist_algo']} to Theorem \ref{['theo:main_theorem']}
  3. Proof of Theorem \ref{['theo:dist_algo']}

Key Result

Theorem 1.2

If $G$ is a graph of finite maximum degree $\Delta$, then $\chi'(G) \leqslant \Delta + 1$.

Theorems & Definitions (14)

  • Definition 1.1: Edge-colorings and chromatic index
  • Theorem 1.2: Vizing Vizing
  • Definition 1.3: Borel graphs and their edge-colorings
  • Theorem 1.4: Kechris--Solecki--Todorcevic KST, Marks Marks
  • Definition 1.5: Subexponential growth
  • Theorem 1.6: Borel Vizing's theorem for subexponential growth graphs
  • Theorem 1.7: Panconesi--Rizzi PR[Theorem 8.5]BE, Chang--He--Li--Pettie--Uitto CHLPU
  • Theorem 1.8: $\mathsf{LOCAL}$ algorithm for Vizing's theorem on subexponential growth graphs
  • Theorem 2.1: AB BernshteynDistributed
  • Definition 3.1: Augmenting subgraphs for partial colorings
  • ...and 4 more