Maximal colourings for graphs

TL;DR

This paper explores two types of graph colourings, introduces their maximum colour counts, and reveals a profound connection between these two notions, advancing understanding in graph theory and spectral analysis.

Contribution

It introduces and compares the maximal colourings for vertices and edges, establishing a deep relationship between these two new graph colouring concepts.

Findings

Defined the t-periodic and periodic colouring numbers.

Established a significant relationship between the two colouring notions.

Provided insights into spectral properties related to these colourings.

Abstract

We consider two different notions of graph colouring, namely, the $t$-periodic colouring for vertices that has been introduced in 1974 by Bondy and Simonovits, and the periodic colouring for oriented edges that has been recently introduced in the context of spectral theory of non-backtracking operators. For each of these two colourings, we introduce the corresponding colouring number which is given by maximising the possible number of colours. We first investigate these two new colouring numbers individually, and we then show that there is a deep relationship between them.