Planar 1-ended graphs can be periodically coloured

TL;DR

This paper proves that planar, 1-ended graphs can always be periodically coloured, with Euclidean cases requiring at most 5 colours, by embedding them into Euclidean or hyperbolic planes and analyzing their symmetries.

Contribution

It establishes the existence of periodic colourings for planar 1-ended graphs and provides a 5-colour bound for Euclidean cases, using geometric embeddings.

Findings

Planar 1-ended graphs admit periodic proper vertex colourings.

Euclidean planar graphs can be coloured with at most 5 colours.

Embedding graphs into Euclidean/hyperbolic planes aids in colouring analysis.

Abstract

We conclude an investigation of Abrishami, Esperet, Giocanti, Hamman, Knappe and M\"oller studying the existence of periodic colourings of locally finite graphs. A colouring of a graph $\Gamma$ is periodic if the resulting coloured graph has a finite number of orbits under its colour-preserving automorphisms, as such it is natural to consider those quasi-transitive graphs with finite quotient. In the case that the graph is planar and has 1-end we prove that it always permits a periodic proper vertex colouring. This is shown by constructing isometry respecting embedded maps into the Euclidean and hyperbolic planes and leveraging known properties of Euclidean and hyperbolic isometry groups. Moreover, in the case that a graph is Euclidean we show that this can always be done in 5 colours.