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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 ,,x m ,x 1 ,,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(c k ) 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:
05C15Coloring of graphs and hypergraphs