$2\times n$ Grids have Unbounded Anagram-Free Chromatic Number

Abstract

We show that anagram-free vertex colouring a $2\times n$ square grid requires a number of colours that increases with $n$. This answers an open question in Wilson's thesis and shows that even graphs of pathwidth $2$ do not have anagram-free colourings with a bounded number of colours.

11.3 pages, 6 figures