Some results in square-free and strong square-free edge-colorings of graphs

The set of problems we consider here are generalizations of square-free sequences [A. Thue, Über unendliche Zeichenreichen, Norske Vid Selsk. Skr. I. Mat. Nat. Kl. Christiana 7 (1906) 1–22]. A finite sequence a1a2 . . . an of symbols from a set S is called square-free if it does not contain a sequence of the form ww = x1x2 . . . xmx1x2 . . . xm, xi ∈ S, as a subsequence of consecutive terms. Extending the above concept to graphs, a coloring of the edge set E in a graph G(V,E) is called square-free if the sequence of colors on any path in G is square-free. This was introduced by Alon et al. [N. Alon, J. Grytczuk, M. Hałuszczak, O. Riordan, Nonrepetitive colorings of graphs, Random Struct.Algor. 21 (2002) 336–346] who proved bounds on the minimum number of colors needed for a square-free edge-coloring of G on the class of graphs with bounded maximum degree and trees. We discuss several variations of this problem and give a few new bounds.