On New Thue Colouring Concepts of Certain Graphs

TL;DR

This paper introduces new concepts in Thue graph colouring, focusing on how edge removal affects the Thue number and proposing a measure for the efficiency of such modifications.

Contribution

It defines the $ au$-index as a novel measure to evaluate the effectiveness of edge deletions in reducing the Thue chromatic number of graphs.

Findings

Analysis of the change in Thue number with edge removal

Introduction of the $ au$-index as an efficiency measure

Insights into graph colouring stability under edge deletions

Abstract

The Thue colouring of a graph is a colouring such that the sequence of vertex colours of any path of even and finite length in $G$ is non-repetitive. The change in the Thue number, $\pi(G)$, as edges are iteratively removed from a graph $G$ is studied. The notion of the $\tau$-index denoted, $\tau(G)$, of a graph $G$ is introduced as well. $\tau(G)$ serves as a measure for the efficiency of edge deletion to reduce the Thue chromatic number of a graph.