## Abstract

An anagram is a word of the form WP where W is a non-empty word and P is a permutation of W. A vertex coloring of a graph is anagram-free if no subpath of the graph is an anagram. Anagram-free graph coloring was independently introduced by Kam\v cev, Luczak, \ and Sudakov [Combin. Probab. Comput., 27 (2018), pp. 623--642] and ourselves [Electron. J. Combin., 25 (2018), pp. 2--20]. In this paper we introduce the study of anagram-free colorings of graph subdivisions. We show that every graph has an anagram-free 8-colorable subdivision. The number of division vertices per edge is exponential in the number of edges. For trees, we construct anagram-free 10-colorable subdivisions with fewer division vertices per edge. Conversely, we prove lower bounds, in terms of division vertices per edge, on the anagram-free chromatic number for subdivisions of the complete graph and subdivisions of complete trees of bounded degree.

Original language | English |
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Pages (from-to) | 2346-2360 |

Number of pages | 15 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 32 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Jan 2018 |

## Keywords

- Anagram-free coloring
- Subdivision
- Trees
- Vertex coloring