Thue choosability of trees Original Research Article

A vertex colouring of a graph image is nonrepetitive if for any path image in image, the first half is coloured differently from the second half. The Thue choice number of image is the least integer image such that for every image-list assignment image of image, there exists a nonrepetitive image-colouring of image. We prove that for any positive integer image, there is a tree image with image. On the other hand, it is proved that if image is a graph of maximum degree image, and image is obtained from image by attaching to each vertex image of image a connected graph of tree-depth at most image rooted at image, then image for some constant image depending only on image and image.