On the chromatic index of path decompositions

If G = (V ′, E) is a graph and H = (V,H) is a graph whose edges can be decomposed into isomorphic copies of G, then we define a k-block colouring of a G−decomposition of H to be an assignment of k colours to the copies of G so that no two copies of G having a vertex in common have the same colour. A G−decomposition of H has chromatic index χ′ = k if it is k block colourable and not k − 1 block colourable. We use the techniques of G−resolvable designs and G−frames to solve the Minimal Chromatic Index Problem: Given H determine mininum {χ′(D) | D is a G−decomposition of H} and exhibit a D that achieves this minimum, for the cases whereH the complete graph on n vertices and G is the path of length 2 or 3.