On T-set of T-colorings

Let image be the collection of sets T such that the greedy algorithm obtains the optimal T-span of image for all image, image the collection of sets T such that image is true for all graphs G. Many families in image or image have been discovered. We know that r-initial set and k multiple of s set are all in image. Liu (T-colorings of graphs, Discrete Math. 10 (1992), 203–212) extended k multiple of s set by union with another set image. In this paper, we continue to study the T-set in image and image, extending some T-sets by the similar way.