# Nonrepetitive edge-colorings of trees

## TL;DR

This paper improves the upper bound on the number of colors needed for nonrepetitive edge-colorings of trees, reducing it from 4Δ-4 to 3Δ-2, thus advancing understanding of graph coloring constraints.

## Contribution

The authors present a simpler nonrepetitive edge-coloring method for trees that improves the upper bound on the Thue edge-chromatic number.

## Key findings

- Established a new upper bound of 3Δ-2 colors for trees.
- Simplified the coloring method compared to previous approaches.
- Enhanced the theoretical understanding of nonrepetitive graph colorings.

## Abstract

A repetition is a sequence of symbols in which the first half is the same as the second half. An edge-coloring of a graph is repetition-free or nonrepetitive if there is no path with a color pattern that is a repetition. The minimum number of colors so that a graph has a nonrepetitive edge-coloring is called its Thue edge-chromatic number. We improve on the best known general upper bound of $4\Delta-4$ for the Thue edge-chromatic number of trees of maximum degree $\Delta$ due to Alon, Grytczuk, Ha{\l}uszczak and Riordan (2002) by providing a simple nonrepetitive edge-coloring with $3\Delta-2$ colors.