Total Thue colourings of graphs

TL;DR

This paper investigates total Thue colourings of graphs, establishing bounds on the minimum number of colours needed for nonrepetitive vertex and edge colourings, with results depending on graph degree and size.

Contribution

It introduces bounds for total Thue and weak total Thue numbers, advancing understanding of nonrepetitive colourings in graphs with new bounds and special family considerations.

Findings

Upper bounds depending on maximum degree and size

Lower bounds for total Thue parameters

Improved bounds for specific graph families

Abstract

A total colouring of a graph is a colouring of its vertices and edges such that no two adjacent vertices or edges have the same colour and moreover, no edge coloured $c$ has its endvertex coloured $c$ too. A weak total Thue colouring of a graph $G$ is a colouring of its vertices and edges such that the colour sequence of consecutive vertices and edges of every path of $G$ is nonrepetitive. In a total Thue colouring also the induced vertex-colouring and edge-colouring of $G$ are nonrepetitive. The weak total Thue number $\pi_{T_w}(G)$ of a graph $G$ denotes the minimum number of colours required in every weak total Thue colouring and the minimum number of colours required in every total Thue colouring is called the total Thue number $\pi_T$. Here we show some upper bounds for both parameters depending on the maximum degree or size of the graph. We also give some lower bounds and some better upper bounds for these graph parameters considering special families of graphs.