The word representation of a graph is a vital study in combinatorics of words and graph theory. This paper defines \(\ell \) -Rauzy graphs (shortly, we say as \(\ell \) -R.G.) for a scattered factorial language. We study the \(\ell \) -R.G. of order k for subwords and scattered subwords of rainbow words and generalized rainbow words. All letters are distinct in rainbow words, and generalized rainbow words are of the form \(a_1^{r_1}a_2^{r_2}\ldots a_s^{r_s}\) . We list and count the vertices and arcs in the \(\ell \) -R.G. for a given word’s factorial language and scattered factorial language. We discuss the connectedness of the \(\ell \) -R.G. for the rainbow word and the generalized rainbow word. We count the isolated vertices in the \(\ell \) -R.G. for the scattered factorial language of a rainbow word.