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On square-free vertex colorings of graphs. (English) Zbl 1164.05021
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.
##### MSC:
 05C15 Coloring of graphs and hypergraphs
##### Keywords:
graph colouring; Thue sequence; outerplanar graph