# Nonrepetitive colourings of graphs excluding a fixed immersion or topological minor

## TL;DR

This paper proves that certain classes of graphs, specifically those excluding a fixed immersion or a particular planar graph as a topological minor, have bounded nonrepetitive chromatic number, expanding understanding of graph coloring constraints.

## Contribution

It establishes the boundedness of nonrepetitive chromatic number for graphs excluding a fixed immersion or a specific planar graph as a topological minor, a new class with this property.

## Key findings

- Graphs excluding a fixed immersion have bounded nonrepetitive chromatic number.
- Graphs excluding a certain planar graph as a topological minor also have bounded nonrepetitive chromatic number.
- This class of graphs is the largest known to have this property.

## Abstract

We prove that graphs excluding a fixed immersion have bounded nonrepetitive chromatic number. More generally, we prove that if $H$ is a fixed planar graph that has a planar embedding with all the vertices with degree at least 4 on a single face, then graphs excluding $H$ as a topological minor have bounded nonrepetitive chromatic number. This is the largest class of graphs known to have bounded nonrepetitive chromatic number.