Repetition-free edge-colorings of k-ary trees
A repetition is a sequence of symbols in which the first half is the same as the second half. An edge-coloring of a graph is repetition-free if there is no path with a color pattern that is a repetition. The minimum number of colors so that a graph has an edge-coloring that is repetition- free is called the Thue edge-chromatic number. In this thesis we investigate the Thue edge-chromatic number of k-ary trees, that is trees in which every vertex has at most k children. Specifically we obtain new upper and lower bounds for the Thue edge-chromatic number of binary trees, and present a new general method for obtaining repetition-free edge-colorings of k-ary trees from what we call k-special sequences. We present examples of k-special sequences as well as algorithms for generating and verifying k-special sequences and repetition-free colorings of k-ary trees.