On New Thue Colouring Concepts of Certain Graphs

Abstract

Let G be a simple undirected graph of order n ≥ 1 and let ϕ be a proper vertex colouring of G. 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, π(G), as edges are iteratively removed from a graph G is studied. Clearly π(G − e) ≤ π(G), e ∈ E(G). As the edges are removed iteratively the resultant values of π are used to construct a Thue sequence for the graph G. The notion of the τ -index denoted, τ (G), of a graph G is introduced as well. τ (G) serves as a measure for the efficiency of edge deletion to reduce the Thue chromatic number of a graph G.