A sequence of symbols is called square-free if it does not contain a subsequence of the form

${x}_{1},\cdots ,{x}_{m},{x}_{1},\cdots ,{x}_{m}$.

*Thue* proved that there is an infinite square-free sequence consisting of three symbols. Sequences can be thought of as colours on the vertices of a path. The authors examine graph colourings for which the colour sequence is square-free on any path. They obtain the result that the vertices of any

$k$-tree have a colouring of this kind using

$O\left({c}^{k}\right)$ colours if

$c>6$. Moreover, they support the conjecture of

*Alon et al.* that a fixed number of colours suffices for any planar graph. They show that this number is at most 12 for outerplanar graphs and construct some outerplanar graphs which require at least 7 colours. Moreover, they construct planar graphs for which at least 10 colours are needed.