Total Thue colourings of graphs
Jens Schreyer, Erika \v{S}krabu\v{l}\'akov\'a

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 has its endvertex coloured too. A weak total Thue colouring of a graph is a colouring of its vertices and edges such that the colour sequence of consecutive vertices and edges of every path of is nonrepetitive. In a total Thue colouring also the induced vertex-colouring and edge-colouring of are nonrepetitive. The weak total Thue number of a graph 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 . 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…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Graph Labeling and Dimension Problems · Advanced Graph Theory Research
