Rainbow cycles through specified vertices

An edge-coloured cycle is rainbow if the edges have distinct colours. Let G be a graph such that any k vertices lie in a cycle of G. The k-rainbow cycle index of G, denoted by crxₖ(G), is the minimum number of colours required to colour the edges of G such that, for every set S of k vertices in G, there exists a rainbow cycle in G containing S. In this paper, we will first prove some results about the parameter crxₖ(G) for general graphs G. One of the results is a classification of all graphs G such that crxₖ(G)=e(G), for $k=1,2$. We will also determine crxₖ(G) for some specific graphs G, including wheels, complete graphs, complete bipartite and multipartite graphs, and discrete cubes.