A note on non-repetitive colourings of planar graphs

TL;DR

This paper explores non-repetitive colourings of planar and outerplanar graphs, establishing upper bounds for face colourings and providing lower bounds for vertex colourings, advancing understanding of graph colouring constraints.

Contribution

It proves that faces of outerplanar maps can be non-repetitively coloured with at most five colours and offers lower bounds for vertex colourings in planar graphs.

Findings

Faces of outerplanar maps are non-repetitively colourable with ≤5 colours

Lower bounds established for vertex non-repetitive colourings in planar graphs

Provides insights into colour requirements for non-repetitive graph colourings

Abstract

Alon et al. introduced the concept of non-repetitive colourings of graphs. Here we address some questions regarding non-repetitive colourings of planar graphs. Specifically, we show that the faces of any outerplanar map can be non-repetitively coloured using at most five colours. We also give some lower bounds for the number of colours required to non-repetitively colour the vertices of both outerplanar and planar graphs.