Anagram-Free Colorings of Graph Subdivisions
Tim E. Wilson, David R. Wood · SIAM Journal on Discrete Mathematics · 2018
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 Kamčev, Łuczak, 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.