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

dThe set of problems we consider here are generalizations of square-free sequences [A. Thue, Uber unendliche Zeichenreichen, Norske Vid Selsk. Skr. 1. Mat. Nat. KL Christiana 7 (1906) 1-22]. A finite sequence a(1)a(2)...a(n) of symbols from a set S is called square-free if it does not contain a sequence of the form ww = x(1)x(2)...x(m)x(1)x(2)...x(m), x(i) is an element of 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. Haluszczak, 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. (C) 2006 Elsevier B.V. All rights reserved.