Anagram-free colorings of graph subdivisions

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 divisi..