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. 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.