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

K.S. Sudeep, Sundar Vishwanathan
2007 Discrete Mathematics  
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 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 = 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
more » ... n 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.
doi:10.1016/j.disc.2006.09.019 fatcat:cxn6zvmxabgejbvgaadtcxwlgi