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

TL;DR

This paper proves that coloring the vertices of a $2\times n$ grid to avoid anagrams requires an increasing number of colors as the grid size grows, answering an open question in graph theory.

Contribution

It demonstrates that even graphs with pathwidth 2 can have unbounded anagram-free chromatic number, resolving a previously open problem.

Findings

Anagram-free coloring number grows with grid size

Graphs of pathwidth 2 do not have bounded anagram-free colorings

Answers an open question in Wilson's thesis

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.