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