Anagram-free colorings of graphs

TL;DR

This paper introduces the concept of anagram-free colorings for graphs, exploring the minimal number of colors needed to prevent anagrammatic sequences along paths, with results on various graph classes including some requiring nearly unique colors.

Contribution

It generalizes the notion of anagram-free sequences to graph colorings and investigates the minimal color requirements for different graph classes, revealing surprising limitations.

Findings

Bounded-degree graphs can require nearly unique colors to avoid anagrams.

The anagram-chromatic number varies significantly across graph classes.

Random regular graphs exemplify cases where many colors are necessary.

Abstract

A sequence $S$ is called anagram-free if it contains no consecutive symbols $r_1 r_2\dots r_k r_{k+1} \dots r_{2k}$ such that $r_{k+1} \dots r_{2k}$ is a permutation of the block $r_1 r_2\dots r_k$. Answering a question of Erd\H{o}s and Brown, Ker\"anen constructed an infinite anagram-free sequence on four symbols. Motivated by the work of Alon, Grytczuk, Ha\l uszczak and Riordan, we consider a natural generalisation of anagram-free sequences for graph colorings. A coloring of the vertices of a given graph $G$ is called anagram-free if the sequence of colors on any path in $G$ is anagram-free. We call the minimal number of colors needed for such a coloring the anagram-chromatic number of $G$. In this paper we study the anagram-chromatic number of several classes of graphs like trees, minor-free graphs and bounded-degree graphs. Surprisingly, we show that there are bounded-degree graphs (such as random regular graphs) in which anagrams cannot be avoided unless we basically give each vertex a separate color.