Anagram-Free Chromatic Number is not Pathwidth-Bounded

TL;DR

This paper demonstrates that the anagram-free chromatic number can be arbitrarily large even for graphs with small pathwidth, challenging previous assumptions about their relationship.

Contribution

It constructs specific planar graphs with small pathwidth that have arbitrarily large anagram-free chromatic numbers, showing no upper bound based on pathwidth.

Findings

Planar graphs of pathwidth 3 can have anagram-free chromatic number growing logarithmically with size.

Graphs with larger pathwidth also exhibit unbounded anagram-free chromatic numbers.

The results challenge prior beliefs about the relationship between graph pathwidth and anagram-free chromatic number.

Abstract

The anagram-free chromatic number is a new graph parameter introduced independently Kam\v{c}ev, {\L}uczak, and Sudakov (2017) and Wilson and Wood (2017). In this note, we show that there are planar graphs of pathwidth 3 with arbitrarily large anagram-free chromatic number. More specifically, we describe $2n$-vertex planar graphs of pathwidth 3 with anagram-free chromatic number $\Omega(\log n)$. We also describe $kn$ vertex graphs with pathwidth $2k-1$ having anagram-free chromatic number in $\Omega(k\log n)$.