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Anagram-Free Colourings of Graphs

Part of: Graph theory

Published online by Cambridge University Press:  08 August 2017

NINA KAMČEV ,
TOMASZ ŁUCZAK  and
BENNY SUDAKOV
NINA KAMČEV
Affiliation:
Department of Mathematics, ETH, Rämistrasse 101, 8092 Zurich, Switzerland (e-mail: nina.kamcev@math.ethz.ch, benjamin.sudakov@math.ethz.ch)
TOMASZ ŁUCZAK
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland (e-mail: tomasz@amu.edu.pl)
BENNY SUDAKOV
Affiliation:
Department of Mathematics, ETH, Rämistrasse 101, 8092 Zurich, Switzerland (e-mail: nina.kamcev@math.ethz.ch, benjamin.sudakov@math.ethz.ch)
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Abstract

A sequence S is called anagram-free if it contains no consecutive symbols r1r2. . .rkrk+1. . .r2k such that rk+1. . .r2k is a permutation of the block r1r2. . .rk. Answering a question of Erdős and Brown, Keränen constructed an infinite anagram-free sequence on four symbols. Motivated by the work of Alon, Grytczuk, Hałuszczak and Riordan [2], we consider a natural generalization of anagram-free sequences for graph colourings. A colouring of the vertices of a given graph G is called anagram-free if the sequence of colours on any path in G is anagram-free. We call the minimal number of colours needed for such a colouring 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 essentially give each vertex a separate colour.

MSC classification

Primary: 05C15: Coloring of graphs and hypergraphs
Secondary: 05C80: Random graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Special Issue 4: Special Issue: Oberwolfach Workshop , July 2018 , pp. 623 - 642
Copyright
Copyright © Cambridge University Press 2017 

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