Anagram-free colorings of graphs
Nina Kam\v{c}ev, Tomasz {\L}uczak, Benny Sudakov

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 is called anagram-free if it contains no consecutive symbols such that is a permutation of the block . 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 is called anagram-free if the sequence of colors on any path in is anagram-free. We call the minimal number of colors needed for such a coloring the anagram-chromatic number of . 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…
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