Borel Vizing's Theorem for Graphs of Subexponential Growth

TL;DR

This paper proves that Borel graphs with subexponential growth can be properly edge-colored with Δ(G)+1 colors, and provides an efficient distributed algorithm for finite graphs of similar growth, extending classical Vizing's theorem.

Contribution

It establishes a Borel version of Vizing's theorem for graphs of subexponential growth and introduces an efficient distributed coloring algorithm for finite such graphs.

Findings

Borel graphs of subexponential growth admit Δ(G)+1 proper edge-colorings.

Finite graphs of subexponential growth can be edge-colored with Δ(G)+1 colors in O(log* n) rounds.

The results depend on the growth rate bounds of the graphs.

Abstract

We show that every Borel graph $G$ of subexponential growth has a Borel proper edge-coloring with $\Delta(G) + 1$ 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 $\Delta(G) + 1$ 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 $G$.