  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 $crx_k(G)$, 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 $crx_k(G)$ for
general graphs $G$. One of the results is a classification of all graphs $G$
such that $crx_k(G)=e(G)$, for $k=1,2$. We will also determine $crx_k(G)$ for
some specific graphs $G$, including wheels, complete graphs, complete bipartite
and multipartite graphs, and discrete cubes.
