# On repetition thresholds of caterpillars and trees of bounded degree

## TL;DR

This paper determines the exact repetition thresholds for caterpillars and degree-3 trees, and provides bounds for thresholds in broader bounded-degree trees, advancing understanding of repetition avoidance in graph-structured words.

## Contribution

It completely characterizes repetition thresholds for caterpillars and degree-3 trees, and offers bounds for other bounded-degree trees, extending the theory of repetition thresholds to graph classes.

## Key findings

- Repetition thresholds for caterpillars are fully determined.
- Repetition thresholds for degree-3 trees are fully determined.
- Bounds are provided for trees with higher bounded degrees.

## Abstract

The repetition threshold is the smallest real number $\alpha$ such that there exists an infinite word over a $k$-letter alphabet that avoids repetition of exponent strictly greater than $\alpha$. This notion can be generalized to graph classes. In this paper, we completely determine the repetition thresholds for caterpillars and caterpillars of maximum degree $3$. Additionally, we present bounds for the repetition thresholds of trees with bounded maximum degrees.