### Abstract

A sequence S is called anagram-free if it contains no consecutive symbols r_{1}r_{2}⋯r_{k}r_{k}+1⋯r_{2k} such that r_{k+1}⋯r_{2k} is a permutation of the block r_{1}r_{2}⋯r_{k}. Answering a question of ErdÅs 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.

Original language | English |
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Pages (from-to) | 623-642 |

Number of pages | 20 |

Journal | Combinatorics Probability and Computing |

Volume | 27 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Jul 2018 |

Externally published | Yes |

## Cite this

*Combinatorics Probability and Computing*,

*27*(4), 623-642. https://doi.org/10.1017/S096354831700027X